Related papers: On Borel sets in ideal topologies

We show that the set of absolutely normal numbers is $\mathbf \Pi^0_3$-complete in the Borel hierarchy of subsets of real numbers. Similarly, the set of absolutely normal numbers is $\Pi^0_3$-complete in the effective Borel hierarchy.

For I a proper, countably complete ideal on P(X) for some set X, can the quotient Boolean algebra P(X)/I be complete? This question was raised by Sikorski in 1949. By a simple projection argument as for measurable cardinals, it can be…

We show that for any uncountable cardinal $\lambda$, the category of sets of cardinality at least $\lambda$ and monomorphisms between them cannot appear as the category of point of a topos, in particular is not the category of models of a…

For an infinite cardinal $\kappa$ let $\ell_2(\kappa)$ be the linear hull of the standard othonormal base of the Hilbert space $\ell_2(\kappa)$ of density $\kappa$. We prove that a non-separable convex subset $X$ of density $\kappa$ in a…

We study the Borel-reducibility of isomorphism relations of complete first order theories and show the consistency of the following: For all such theories T and T', if T is classifiable and T' is not, then the isomorphism of models of T' is…

An ideal on a set $X$ is a collection of subsets of $X$ closed under the operations of taking finite unions and subsets of its elements. Ideals are a very useful notion in topology and set theory and have been studied for a long time. We…

We prove that topological isomorphism on procountable groups is not classifiable by countable structures, in the sense of descriptive set theory. In fact, the equivalence relation $\ell_\infty$ expressing that two sequences of reals have a…

We investigate generalizations of the topology of the higher Cantor space on $2^\kappa$, based on arbitrary ideals rather than the bounded ideal on $\kappa$. Our main focus is on the topology induced by the nonstationary ideal, and we call…

We generalize the basic theory of universally Baire sets of $2^\omega$ to a theory of universally Baire subsets of $2^\kappa$. We show that the fundamental characterizations of the property of being universally Baire have natural…

We show that if A is a linear order then Th(A) is either $\aleph_0$-categorical or Borel complete (in the sense of Friedman and Stanley). We generalize this; if A has countably many unary predicates attached, then Th(A) is…

The property of countable metacompactness of a topological space gets its importance from Dowker's 1951 theorem that the product of a normal space X with the unit interval is again normal iff X is countably metacompact. In a recent paper,…

We present the effective version of the theorem about turning Borel sets in Polish spaces into clopen sets while preserving the Borel structure of the underlying space. We show that under some conditions the emerging parameters can be…

We study aleph_0-stable theories, and prove that if T either has eni-DOP or is eni-deep, then its class of countable models is Borel complete. We introduce the notion of lambda-Borel completeness and prove that such theories are…

The cut pseudo-metric on the space of graph limits induces an equivalence relation. The quotient space obtained by collapsing each equivalence class to a point is a metric space with appealing analytic properties. We show that the…

A topological space $X$ is said to be an Ascoli space if any compact subset $K$ of $C_k(X)$ is evenly continuous. This definition is motivated by the classical Ascoli theorem. We study the $k_R$-property and the Ascoli property of…

We determine the exact Borel class of the points whose iterates under $\exp(z)+a$ tend to infinity. We also prove that the sets of non-escaping Julia points for many of these functions are topologically equivalent.

We provide a comprehensive development of the basics of descriptive set theory for non-separable complete metric spaces whose weight is a singular cardinal $\lambda$ of countable confinality. Somewhat unexpectedly, the resulting theory is…

We prove under $V=L$ that the inclusion modulo the non-stationary ideal is a $\Sigma_1^1$-complete quasi-order in the generalized Borel-reducibility hierarchy ($\kappa>\omega$). This improvement to known results in $L$ has many new…

It is consistent that the continuum be arbitrary large and no absolute $\kappa$-Borel set $X$ of density $\kappa$, $\aleph_1<\kappa<\mathfrak{c}$, condenses onto a compact metric space. It is consistent that the continuum be arbitrary large…

We prove that the countable product of lines contains a Borel linear subspace $L\ne\mathbb R^\omega$ that cannot be covered by countably many closed Haar-meager sets. This example is applied to studying the interplay between various classes…