arXiv subjects

Claudio Agostini

Publications and source records attributed to Claudio Agostini.

9 recordsLinked to original sources

arXiv2026

On the problem of generalized measures: an impossibility result

This paper investigates the problem of extending measure theory to non-separable structures, from generalized descriptive set theory to a broader class of spaces beyond this framework. While various notions, such as the ideal of measure zero sets, have been generalized, the question of whether a satisfactory notion of $\lambda^+$-measure could be defined in generalized descriptive set theory has remained open. We introduce a broad class of $\lambda^+$-measures as functions taking values in arbitrary positively totally ordered monoids equipped with an infinitary sum. This definition relies on minimal assumptions and captures most natural generalizations of measures to this context. We then prove that, under certain cardinal assumptions, no continuous $\lambda^+$-measure of this kind exists on ${}^\kappa\lambda$, nor on any $\lambda^+$-Borel space or $T_0$ topological space of weight at most $\lambda$. We also show the optimality of these cardinal assumptions.

math.LO↗

arXiv2025

Generalized Borel Sets

Generalizing classical descriptive set theory opens foundational questions about the Borel hierarchy. In this paper we systematically study those questions, working in the general framework of Polish-like spaces relative to an uncountable cardinal $\kappa$, possibly singular, satisfying $2^{<\kappa}=\kappa$. We provide fundamental properties of the $\kappa^+$-Borel hierarchy of any regular Hausdorff space of weight at most $\kappa$, and establish sufficient conditions for its non-collapse. We highlight a unique phenomenon that arises in the case of singular cardinals, namely, the existence of a second, distinct Borel hierarchy, the $\kappa$-Borel hierarchy: we prove that it is strictly finer than the $\kappa^+$-Borel hierarchy, and then characterize the precise relationship between the two. Finally, for regular cardinals, we resolve three questions about the behavior of the $\kappa^+$-Borel hierarchy on subspaces of the generalized Baire space ${}^\kappa \kappa$, constructing various models via forcing where several nontrivial constellations for the length of the $\kappa^+$-Borel hierarchy on the space are realized.

math.LO↗

arXiv2025

Countable dense homogeneity and topological groups

Building on results of Medvedev, we construct a $\mathsf{ZFC}$ example of a non-Polish topological group that is countable dense homogeneous. Our example is a dense subgroup of $\mathbb{Z}^\omega$ of size $\mathfrak{b}$ that is a $\lambda$-set. We also conjecture that every countable dense homogenous Baire topological group with no isolated points contains a copy of the Cantor set, and give a proof in a very special case.

math.GN↗

arXiv2024

Every finite-dimensional analytic space is $\sigma$-homogeneous

All spaces are assumed to be separable and metrizable. Building on work of van Engelen, Harrington, Michalewski and Ostrovsky, we obtain the following results: (1) Every finite-dimensional analytic space is $\sigma$-homogeneous with analytic witnesses, (2) Every finite-dimensional analytic space is $\sigma$-homogeneous with pairwise disjoint $\mathbf{\Delta}^1_2$ witnesses. Furthermore, the complexity of the witnesses is optimal in both of the above results. This completes the picture regarding $\sigma$-homogeneity in the finite-dimensional realm. It is an open problem whether every analytic space is $\sigma$-homogeneous. We also investigate finite unions of homogeneous spaces.

math.GN↗

arXiv2024

Continuous logic in a classical setting

Let $\mathcal{L}$ be a first-order two-sorted language and consider a class of $\mathcal{L}$-structures of the form $\langle M, X \rangle$ where $M$ varies among structures of the first sort, while $X$ is fixed in the second sort, and it is assumed to be a compact Hausdorff space. When $X$ is a compact subset of the real line, one way to treat classes of this kind model-theoretically is via continuous-valued logic, as in [Ben Yaacov-Berenstein-Henson-Usvyatsov 2010]. Prior to that, Henson and Iovino proposed an approach based on the notion of positive formulas [Henson-Iovino 2002]. Their work is tailored to the model theory of Banach spaces. Here we show that a similar approach is possible for a more general class of models. We introduce suitable versions of elementarity, compactness, saturation, quantifier elimination and other basic tools, and we develop basic model theory.

math.LO↗

arXiv2023

Countable spaces, realcompactness, and the pseudointersection number

All spaces are assumed to be Tychonoff. Given a realcompact space $X$, we denote by $\mathsf{Exp}(X)$ the smallest infinite cardinal $\kappa$ such that $X$ is homeomorphic to a closed subspace of $\mathbb{R}^\kappa$. Our main result shows that, given a cardinal $\kappa$, the following conditions are equivalent: $(1)$ There exists a countable crowded space $X$ such that $\mathsf{Exp}(X)=\kappa$, $(2)$ $\mathfrak{p}\leq\kappa\leq\mathfrak{c}$. In fact, in the case $\mathfrak{d}\leq\kappa\leq\mathfrak{c}$, every countable dense subspace of $2^\kappa$ provides such an example. This will follow from our analysis of the pseudocharacter of countable subsets of products of first-countable spaces. Finally, we show that a scattered space of weight $\kappa$ has pseudocharacter at most $\kappa$ in any compactification. This will allow us to calculate $\mathsf{Exp}(X)$ for an arbitrary (that is, not necessarily crowded) countable space.

math.GN↗

arXiv2023

Generalized Polish spaces at regular uncountable cardinals

In the context of generalized descriptive set theory, we systematically compare and analyze various notions of Polish-like spaces and standard $κ$-Borel spaces for $κ$ an uncountable (regular) cardinal satisfying $κ^{<κ} = κ$. As a result, we obtain a solid framework where one can develop the theory in full generality. We also provide natural characterizations of the generalized Cantor and Baire spaces. Some of the results obtained considerably extend previous work from [Coskey-Schlicht 2016, Galeotti 2019, Luecke-Schlicht 2015], and answer some questions contained therein.

math.LO↗

arXiv2021

Ramsey monoids

Recently, Solecki introduced the notion of Ramsey monoid to produce a common generalization to theorems such as Hindman's theorem, Carlson's theorem, and Gowers' FIN$_k$ theorem. He proved that an entire class of finite monoids is Ramsey. Here we improve this result, enlarging this class and finding a simple algebraic characterization of finite Ramsey monoids. We extend in a similar way a result of Solecki regarding a second class of monoids connected to the Furstenberg-Katznelson Ramsey Theorem. The results obtained suggest a possible connection with Schützenberger's theorem and finite automata theory.

math.CO↗

arXiv2018

On bisequentiality and spaces of strictly decreasing functions on trees

We present a characterization of spaces of strictly decreasing functions on trees in terms of bisequentiality. This characterization answers Questions 6.1 and 6.2 of "A filter on a collection of finite sets and Eberlein compacta" by T. Cieśla. Moreover we study the relation between these spaces and the classes of Corson, Eberlein and uniform Eberlein compacta.

math.GN↗