We give several refinements of known theorems on Borel uniformizations of sets with “large sections”. In particular, we show that a set B ⊂ [0,1] × [0,1] which belongs to $\Sigma {⁰}_{\alpha }$, α ≥ 2, and which has all “vertical” sections of positive Lebesgue measure, has a $\Pi {⁰}_{\alpha }$ uniformization which is the graph of a $\Sigma {⁰}_{\alpha }$-measurable mapping. We get a similar result for sets with nonmeager sections. As a corollary we derive an improvement of Srivastava’s theorem on uniformizations for Borel sets with $G_{\delta }$ sections. ...

We prove that if a Δ¹₁ function f with Σ¹₁ domain X is σ-continuous then one can find a Δ¹₁ covering ${\left(Aₙ\right)}_{n\in \omega }$ of X such that $f_{|Aₙ}$ is continuous for all n. This is an effective version of a recent result by Pawlikowski and Sabok, generalizing an earlier result of Solecki.

Let (X,τ) be a countable topological space. We say that τ is an analytic (resp. Borel) topology if τ as a subset of the Cantor set $2^X$ (via characteristic functions) is an analytic (resp. Borel) set. For example, the topology of the Arkhangel’skiĭ-Franklin space $S_{\omega }$ is $F_{\sigma \delta }$. In this paper we study the complexity, in the sense of the Borel hierarchy, of subspaces of $S_{\omega }$. We show that $S_{\omega }$ has subspaces with topologies of arbitrarily high Borel rank and it also has subspaces with a non-Borel topology....

Jayne and Rogers proved that every function from an analytic space into a separable metrizable space is decomposable into countably many continuous functions with closed domains if and only if the preimage of each $F_{\sigma }$ set under that function is again $F_{\sigma }$. Many researchers conjectured that the Jayne-Rogers theorem can be generalized to all finite levels of Borel functions. In this paper, by using the Shore-Slaman join theorem on the Turing degrees, we show the following variant of the Jayne-Rogers...

In every infinite-dimensional Fréchet space X, we construct a linear subspace E such that E is an $F_{\sigma \delta \sigma }$-subset of X and contains a retract R so that $R\times E^{\omega }$ is not homeomorphic to $E^{\omega }$. This shows that Toruńczyk’s Factor Theorem fails in the Borel case.

By Fin(X) (resp. $Fi{n}^k\left(X\right)$), we denote the hyperspace of all non-empty finite subsets of X (resp. consisting of at most k points) with the Vietoris topology. Let ℓ₂(τ) be the Hilbert space with weight τ and $\ell {₂}^f\left(\tau \right)$ the linear span of the canonical orthonormal basis of ℓ₂(τ). It is shown that if $E=\ell {₂}^f\left(\tau \right)$ or E is an absorbing set in ℓ₂(τ) for one of the absolute Borel classes ${}_{\alpha }\left(\tau \right)$ and ${}_{\alpha }\left(\tau \right)$ of weight ≤ τ (α > 0) then Fin(E) and each $Fi{n}^k\left(E\right)$ are homeomorphic to E. More generally, if X is a connected E-manifold then Fin(X)...

For each ordinal 1 ≤ α < ω₁ we present separable metrizable spaces $X_{\alpha }$, $Y_{\alpha }$ and $Z_{\alpha }$ such that (i) $f{X}_{\alpha },f{Y}_{\alpha },f{Z}_{\alpha }=\omega ₀$, where f is either trdef or ₀-trsur, (ii) $A\left(\alpha \right)-trind{X}_{\alpha }=\infty$ and $M\left(\alpha \right)-trind{X}_{\alpha }=-1$, (iii) $A\left(\alpha \right)-trind{Y}_{\alpha }=-1$ and $M\left(\alpha \right)-trind{Y}_{\alpha }=\infty$, and (iv) $A\left(\alpha \right)-trind{Z}_{\alpha }=M\left(\alpha \right)-trind{Z}_{\alpha }=\infty$ and $A(\alpha +1)\cap M(\alpha +1)-trind{Z}_{\alpha }=-1$. We also show that there exists no separable metrizable space $W_{\alpha }$ with $A\left(\alpha \right)-trind{W}_{\alpha }\ne \infty$, $M\left(\alpha \right)-trind{W}_{\alpha }\ne \infty$ and $A\left(\alpha \right)\cap M\left(\alpha \right)-trind{W}_{\alpha }=\infty$, where A(α) (resp. M(α)) is the absolutely additive (resp. multiplicative) Borel class.

It is proved that $F_{\sigma }$-mappings preserve absolute Borel classes, which improves results of R. W. Hansell, J. E. Jayne and C. A. Rogers. The proof is based on the fact that any $F_{\sigma }$-mapping f: X → Y of an absolute Suslin metric space X onto an absolute Suslin metric space Y becomes a piecewise perfect mapping when restricted to a suitable $F_{\sigma }$-set $X_{\infty }\subset X$ satisfying $f\left(X_{\infty }\right)=Y$.

An example of a non-zero non-atomic translation-invariant Borel measure ${\nu }_p$ on the Banach space ${\ell }_p\left(1\le p\le \infty \right)$ is constructed in Solovay’s model. It is established that, for 1 ≤ p < ∞, the condition "${\nu }_p$-almost every element of ${\ell }_p$ has a property P" implies that “almost every” element of ${\ell }_p$ (in the sense of [4]) has the property P. It is also shown that the converse is not valid.

For a linear operator T in a Banach space let ${\sigma }_p\left(T\right)$ denote the point spectrum of T, let ${\sigma }_{p,n}\left(T\right)$ for finite n > 0 be the set of all $\lambda \in {\sigma }_p\left(T\right)$ such that dim ker(T - λ) = n and let ${\sigma }_{p,\infty }\left(T\right)$ be the set of all $\lambda \in {\sigma }_p\left(T\right)$ for which ker(T - λ) is infinite-dimensional. It is shown that ${\sigma }_p\left(T\right)$ is ${ℱ}_{\sigma }$, ${\sigma }_{p,\infty }\left(T\right)$ is ${ℱ}_{\sigma \delta }$ and for each finite n the set ${\sigma }_{p,n}\left(T\right)$ is the intersection of an ${ℱ}_{\sigma }$ set and a ${}_{\delta }$ set provided T is closable and the domain of T is separable and weakly σ-compact. For closed densely defined operators in a separable Hilbert space a more...

Let X be a Polish space, and let C₀ and C₁ be disjoint coanalytic subsets of X. The pair (C₀,C₁) is said to be complete if for every pair (D₀,D₁) of disjoint coanalytic subsets of ${\omega }^{\omega }$ there exists a continuous function $f:{\omega }^{\omega }\to X$ such that $f^{-1}\left(C₀\right)=D₀$ and $f^{-1}\left(C₁\right)=D₁$. We give several explicit examples of complete pairs of coanalytic sets.

Let (ₙ)ₙ be a quasianalytic differentiable system. Let m ∈ ℕ. We consider the following problem: let $f{\in }_m$ and f̂ be its Taylor series at $0\in {ℝ}^m$. Split the set ${\mathbb{N}}^m$ of exponents into two disjoint subsets A and B, ${\mathbb{N}}^m=A\cup B$, and decompose the formal series f̂ into the sum of two formal series G and H, supported by A and B, respectively. Do there exist $g,h{\in }_m$ with Taylor series at zero G and H, respectively? The main result of this paper is the following: if we have a positive answer to the above problem for some...

Let $K=\mathbb{Q}\left(\alpha \right)$ be a number field generated by a complex root $\alpha$ of a monic irreducible polynomial $f\left(x\right)=x^{18}-m$, $m\ne \mp 1$, is a square free rational integer. We prove that if $m\equiv 2$ or $3(\mathrm{mod}4)$ and $m\neg \equiv \mp 1(\mathrm{mod}9)$, then the number field $K$ is monogenic. If $m\equiv 1(\mathrm{mod}4)$ or $m\equiv 1(\mathrm{mod}9)$, then the number field $K$ is not monogenic.

Let P and Q be nonzero integers. The sequences of generalized Fibonacci and Lucas numbers are defined by U₀ = 0, U₁ = 1 and $U_{n+1}=PUₙ-Q{U}_{n-1}$ for n ≥ 1, and V₀ = 2, V₁ = P and $V_{n+1}=PVₙ-Q{V}_{n-1}$ for n ≥ 1, respectively. In this paper, we assume that P ≥ 1, Q is odd, (P,Q) = 1, Vₘ ≠ 1, and $V_r\ne 1$. We show that there is no integer x such that $Vₙ=V_{r}Vₘx²$ when m ≥ 1 and r is an even integer. Also we completely solve the equation $Vₙ=VₘV_{r}x²$ for m ≥ 1 and r ≥ 1 when Q ≡ 7 (mod 8) and x is an even integer. Then we show that when P ≡ 3 (mod 4) and...

We study the generalized Cantor space ${}^{\kappa }2$ and the generalized Baire space ${}^{\kappa }\kappa$ as analogues of the classical Cantor and Baire spaces. We equip ${}^{\kappa }\kappa$ with the topology where a basic neighborhood of a point η is the set ν: (∀j < i)(ν(j) = η(j)), where i < κ. We define the concept of a strong measure zero set of ${}^{\kappa }2$. We prove for successor $\kappa ={\kappa }^{<\kappa }$ that the ideal of strong measure zero sets of ${}^{\kappa }2$ is ${}_{\kappa }$-additive, where ${}_{\kappa }$ is the size of the smallest unbounded family in ${}^{\kappa }\kappa$, and that the generalized Borel...

Applying results of the infinitary Ramsey theory, namely the dichotomy principle of Galvin-Prikry, we show that for every sequence ${\left({\alpha }_j\right)}_{j=1}^{\infty }$ of scalars, there exists a subsequence ${\left({\alpha }_{k_j}\right)}_{j=1}^{\infty }$ such that either every subsequence of ${\left({\alpha }_{k_j}\right)}_{j=1}^{\infty }$ defines a universal series, or no subsequence of ${\left({\alpha }_{k_j}\right)}_{j=1}^{\infty }$ defines a universal series. In particular examples we decide which of the two cases holds.

Given a topological space ⟨X,⟩ ∈ M, an elementary submodel of set theory, we define $X_M$ to be X ∩ M with topology generated by $U\cap M:U\in \cap M$. Suppose $X_M$ is homeomorphic to the irrationals; must $X=X_M$? We have partial results. We also answer a question of Gruenhage by showing that if $X_M$ is homeomorphic to the “Long Cantor Set”, then $X=X_M$.

Let $H\subseteq Z\subseteq 2^{\omega }$. For n ≥ 2, we prove that if Selivanovski measurable functions from $2^{\omega }$ to Z give as preimages of H all Σₙ¹ subsets of $2^{\omega }$, then so do continuous injections.

Let β ∈ (1,2) and x ∈ [0,1/(β-1)]. We call a sequence ${\left({ϵ}_i\right)}_{i=1}^{\infty }\in {0,1}^{\mathbb{N}}$ a β-expansion for x if $x={\sum }_{i=1}^{\infty }{ϵ}_i{\beta }^{-i}$. We call a finite sequence ${\left({ϵ}_i\right)}_{i=1}^n\in {0,1}^n$ an n-prefix for x if it can be extended to form a β-expansion of x. In this paper we study how good an approximation is provided by the set of n-prefixes. Given $\Psi :\mathbb{N}\to {ℝ}_{\ge 0}$, we introduce the following subset of ℝ: $W_{\beta }\left(\Psi \right):={\bigcap }_{m=1}^{\infty }{\bigcup }_{n=m}^{\infty }{\bigcup }_{{\left({ϵ}_i\right)}_{i=1}^n\in {0,1}^n}[{\sum }_{i=1}^n\left({ϵ}_i\right)/\left({\beta }^i\right),{\sum }_{i=1}^n\left({ϵ}_i\right)/\left({\beta }^i\right)+\Psi \left(n\right)]$In other words, $W_{\beta }\left(\Psi \right)$ is the set of x ∈ ℝ for which there exist infinitely many solutions to the inequalities $0\le x-{\sum }_{i=1}^n\left({ϵ}_i\right)/\left({\beta }^i\right)\le \Psi \left(n\right)$. When ${\sum }_{n=1}^{\infty }{2}^n\Psi \left(n\right)<\infty$, the Borel-Cantelli lemma tells us that the Lebesgue measure...

Let ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to d}$ be the following algorithmic problem: Given a finite simplicial complex $K$ of dimension at most $k$, does there exist a (piecewise linear) embedding of $K$ into ${ℝ}^d$? Known results easily imply polynomiality of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to 2}$ ($k=1,2$; the case $k=1,d=2$ is graph planarity) and of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to 2k}$ for all $k\ge 3$. We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{d\to d}$ and ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{(d-1)\to d}$ are undecidable for each ${}_d\ge 5$. Our main result is NP-hardness of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{2\to 4}$ and, more generally, of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to d}$ for all...

We show that any ${\Sigma }_s$-product of at most $𝔠$-many $L\Sigma (\le \omega )$-spaces has the $L\Sigma (\le \omega )$-property. This result generalizes some known results about $L\Sigma (\le \omega )$-spaces. On the other hand, we prove that every ${\Sigma }_s$-product of monotonically monolithic spaces is monotonically monolithic, and in a similar form, we show that every ${\Sigma }_s$-product of Collins-Roscoe spaces has the Collins-Roscoe property. These results generalize some known results about the Collins-Roscoe spaces and answer some questions due to Tkachuk [Lifting the Collins-Roscoe...

We investigate the question whether a system ${\left(E_i\right)}_{i\in I}$ of homogeneous linear equations over $\mathbb{Z}$ is non-trivially solvable in $\mathbb{Z}$ provided that each subsystem ${\left(E_j\right)}_{j\in J}$ with $\left|J\right|\le c$ is non-trivially solvable in $\mathbb{Z}$ where $c$ is a fixed cardinal number such that $c<\left|I\right|$. Among other results, we establish the following. (a) The answer is ‘No’ in the finite case (i.e., $I$ being finite). (b) The answer is ‘No’ in the denumerable case (i.e., $\left|I\right|={\aleph }_0$ and $c$ a natural number). (c) The answer in case that $I$ is uncountable and $c\le {\aleph }_0$ is ‘No...

Let ${\alpha }_i,{\beta }_i>0$, 1 ≤ i ≤ n, and for t > 0 and x = (x₁,...,xₙ) ∈ ℝⁿ, let $t•x=(t^{\alpha ₁}x₁,...,t^{\alpha ₙ}xₙ)$, $t\circ x=(t^{\beta ₁}x₁,...,t^{\beta ₙ}xₙ)$ and $\left|\right|x\left|\right|={\sum }_{i=1}^n{\left|x_i\right|}^{1/{\alpha }_i}$. Let φ₁,...,φₙ be real functions in $C^{\infty }(ℝⁿ-0)$ such that φ = (φ₁,..., φₙ) satisfies φ(t • x) = t ∘ φ(x). Let γ > 0 and let μ be the Borel measure on ${ℝ}^{2n}$ given by $\mu \left(E\right)={\int }_{ℝⁿ}{\chi }_E{(x,\phi \left(x\right))\left|\right|x\left|\right|}^{\gamma -\alpha }dx$, where $\alpha ={\sum }_{i=1}^n{\alpha }_i$ and dx denotes the Lebesgue measure on ℝⁿ. Let $T_{\mu }f=\mu \ast f$ and let $\left|\right|T_{\mu }{\left|\right|}_{p,q}$ be the operator norm of $T_{\mu }$ from $L^p\left({ℝ}^{2n}\right)$ into $L^q\left({ℝ}^{2n}\right)$, where the $L^p$ spaces are taken with respect to the Lebesgue measure. The type set $E_{\mu }$ is defined by $E_{\mu }=(1/p,1/q):\left|\right|T_{\mu }{\left|\right|}_{p,q}<\infty ,1\le p,q\le \infty$. In the case ${\alpha }_i\ne {\beta }_k$ for 1 ≤ i,k ≤ n we characterize the...