Related papers: Borel* Sets in the Generalised Baire Space

We prove that for any homogeneous structure $\mathbf{K}$ in a language with finitely many relation symbols of arity at most two satisfying SDAP$^+$ (or LSDAP$^+$), there are spaces of subcopies of $\mathbf{K}$, forming subspaces of the…

We give a survey of cardinal charcteristics of the higher Cicho\'n diagram defined on the higher Baire space ${}^\kappa\kappa$ for $\kappa$ regular with $2^{<\kappa}=\kappa$. Specifically, we will compare consistency proofs from the…

We generalize a construction of Barthel-Brasselet-Fieseler-Gabber-Kaup in the setting of complex varieties to the setting of finite type, complex algebraic stacks. Given two such stacks $\mathcal{X},\mathcal{Y}$ with affine stabilizers, and…

We introduce the open degree of a compact space, and we show that for every natural number n, the separable Rosenthal compact spaces of degree n have a finite basis.

We study the Borel complexity of sets of normal numbers in several numeration systems. Taking a dynamical point of view, we offer a unified treatment for continued fraction expansions and base $r$ expansions, and their various…

The paper considers the spaces $B_p[1, \alpha]$ of all Baire functions $x\colon [1,\alpha]\to \mathbb{R}$, defined on segments of ordinals $[1,\alpha]$ and endowed with the topology of pointwise convergence. A complete topological…

We prove the Borel Conjecture for a class of groups containing word-hyperbolic groups and groups acting properly, isometrically and cocompactly on a finite dimensional CAT(0)-space.

For each subset of Baire space, we define, in away similar to a common proof of the Cantor-Bendixson Theorem, a sequence of decreasing subsets S_alpha of N^N, indexed by ordinals. We use this to obtain two new characterizations of the…

The Borel map takes a smooth function to its infinite jet of derivatives (at zero). We study the restriction of this map to ultradifferentiable classes of Beurling type in a very general setting which encompasses the classical…

We prove several theorems on sigma-bounded and sigma-compact pointsets. We start with a known theorem by Kechris, saying that any lightface \Sigma^1_1 set of the Baire space either is effectively sigma-bounded (that is, covered by a…

We show that, consistently, there exists a Borel set B subset Cantor admitting a sequence (eta_alpha:alpha<lambda) of distinct elements of Cantor such that (eta_alpha+B) cap (eta_beta+B) is uncountable for all alpha,beta<lambda but with no…

We provide a finite basis for the class of Borel functions that are not in the first Baire class, as well as the class of Borel functions that are not $\sigma$-continuous with closed witnesses.

This paper answers three questions posed by the first author. In Theorem 2.6 we show that the family of strong measure zero subsets of {}^{omega_1}2 is 2^{aleph_1}-additive under GMA and CH. In Theorem 3.1 we prove that the generalized…

Let $\Gamma^\infty$ be the set of all universally Baire sets of reals. Inspired by recent work of the second author and Nam Trang, we introduce a new technique for establishing generic absoluteness results for models containing…

Say that a cardinal number $\kappa$ is \emph{small} relative to the space $X$ if $\kappa <\Delta(X)$, where $\Delta(X)$ 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…

Working with uncountable structures of fixed cardinality, we investigate the complexity of certain equivalence relations and show that if V = L, then many of them are \Sigma^1_1-complete, in particular the isomorphism relation of dense…

We generalise the main theorems from the paper "The Borel cardinality of Lascar strong types" by I. Kaplan, B. Miller and P. Simon to a wider class of bounded invariant equivalence relations. We apply them to describe relationships between…

The following will be shown: Let $I$ be a $\sigma$-ideal on a Polish space $X$ with the property that the associated forcing of $I^+$ Borel subsets ordered by $\subseteq$ is a proper forcing. Let E be an analytic or coanalytic equivalence…

Using Koszmider's strongly unbounded functions, we show the following consistency result: Suppose that $\kappa,\lambda$ are infinite cardinals such that $\kappa^{+++} \leq \lambda$, $\kappa^{<\kappa}=\kappa$ and $2^{\kappa}= \kappa^+$, and…

Let $X$ be an uncountable Polish space and let $\mathcal{I}$ be an ideal on $\omega$. A point $\eta \in X$ is an $\mathcal{I}$-limit point of a sequence $(x_n)$ taking values in $X$ if there exists a subsequence $(x_{k_n})$ convergent to…