Related papers: On extendability to $F_\sigma$ ideals

Answering a question of Hru\v{s}\'ak, we show that every analytic tall ideal on $\omega$ contains an $F_\sigma$ tall ideal. We also give an example of an $F_\sigma$ tall ideal without a Borel selector.

We address some phenomena about the interaction between lower semicontinuous submeasures on $\mathbb{N}$ and $F_{\sigma}$ ideals. We analyze the pathology degree of a submeasure and present a method to construct pathological $F_\sigma$…

For any abelian Polish sigma-compact group H there exist a sigma-ideal Z over N and a Borel Z-approximate homomorphism f : H --> H^N which is not Z-approximable by a continuous true homomorphism g : H --> H^N.

Answering a question of the second listed author we show that there is no tall Borel ideal minimal among all tall Borel ideals in the Kat\v{e}tov order.

We study Egorov ideals, that is ideals on $\omega$ for which the Egorov's theorem for ideal versions of pointwise and uniform convergences holds. We show that a non-pathological $\bf{\Sigma^0_2}$ ideal is Egorov if and only if it is…

Given an analytic equivalence relation, we tend to wonder whether it is Borel. When it is non Borel, there is always the hope it will be Borel on a "large" set -- nonmeager or of positive measure. That has led Kanovei, Sabok and Zapletal to…

Let $X$ be a zero-dimensional compact metrizable space endowed with a strictly positive continuous Borel $\sigma$-additive measure $\mu$ which is good in the sense that for any clopen subsets $U,V\subset X$ with $\mu(U)<\mu(V)$ there is a…

Following the topic of the book Canonical Ramsey Theory on Polish Spaces by V. Kanovei, M. Sabok and J. Zapletal we study Borel equivalences on Laver trees. Here we prove that equivalence relations Borel reducible to an equivalence relation…

We provide a counterexample to the Category Dichotomy in the framework of $\textsf{ZFC}$. That is, we prove the existence of an ideal on $\omega$ that is not Kat\v{e}tov below $\mathsf{nwd}$ and does not have restrictions above…

We give necessary and sufficient conditions on an Ore extension $A[x;\sigma,\delta]$, where $A$ is a finite dimensional algebra over a field $\mathbb{F}$, for being a Frobenius extension over the ring of commutative polynomials…

We show that the Nash-Williams theorem has a uniform version and that the Galvin theorem does not. We show that there is an $F_\sigma$ tall ideal on $\mathbb{N}$ without a Borel selector and also construct a $\mathbf\Pi^1_2$ tall ideal…

We investigate the possibility of extension of $F_\sigma$-measurable and Baire-one maps from subspaces of topological spaces when these maps take values in spaces which covers by a sequence of metrizable spaces with special properties

We introduce a method that allows to turn topological questions about Hindman spaces into purely combinatorial questions about the Kat\v{e}tov order of ideals on $\mathbb{N}$. We also provide two applications of the method. (1) We…

Let $\mu$ be a Borel measure on a compactum $X$. The main objects in this paper are $\sigma$-ideals $I(dim)$, $J_0(\mu)$, $J_f(\mu)$ of Borel sets in $X$ that can be covered by countably many compacta which are finite-dimensional, or of…

We study and classify topologically invariant $\sigma$-ideals with a Borel base on the Hilbert cube and evaluate their cardinal characteristics. One of the results of this paper solves (positively) a known problem whether the minimal…

In this paper we shall consider a couple of properties of $\sigma$-ideals and study relations between them. Namely we will prove that $\mathfrak{c}$-cc $\sigma$-ideals are tall and that the Weaker Smital Property implies that every Borel…

Let $F_{\omega_1}$ be the countable admissible ordinal equivalence relation defined on ${}^\omega 2$ by $x \ F_{\omega_1} \ y$ if and only if $\omega_1^x = \omega_1^y$. It will be shown that $F_{\omega_1}$ is classifiable by countable…

In this paper, we shall discuss the extendability of probability and non-probability measures on Cayley trees to a $\sigma$-additive measure on Borel fields which has a fundamental role in the theory of Gibbs measures.

We show that every basis for the countable Borel equivalence relations strictly above $\mathbb{E}_0$ under measure reducibility is uncountable, thereby ruling out natural generalizations of the Glimm-Effros dichotomy. We also push many…

Answering a question of Elekes and Vidny\'anszky, we construct a Polish meta-abelian group $H$ and a subgroup $F\subset H$, which is a Haar null $F_\sigma$-set in $H$ that cannot be enlarged to a Haar null $G_\delta$-set.