Schedule for: 23w5025 - Set-Theoretic Topology

The Borel partition spectrum is the set of all uncountable cardinals κ such that there is a partition of the real line into precisely κ Borel sets. We will look at four theorems concerning this spectrum: three of them provable from ZFC, and one provable from ZFC + "0-dagger does not exist". I will then outline a proof that any set of cardinals not violating these four requirements can be made into the Borel partition spectrum in a forcing extension. Several open questions are included.

$mH$- and $H$-separability are two topological selection principles modifying the notion of selective separability introduced in work by Bella, Bonanzinga, and Matveev. Selective separability was introduced by Scheepers using the well-known selection principle $S_{fin}(\mathcal A, \mathcal B)$. It is known that countable Fréchet-Urysohn spaces are $mH$-separable. Recent work of Bardyla, Maesano, and Zdomskyy shows that $\mathfrak p = \mathfrak c$ implies there is a countable Fréchet-Urysohn space that is not $H$-separable. We construct in $ZFC$ another example of a countable space which distinguishes $mH$ from $H$ using a transfinite recursion of length $\mathfrak b$. Additionally, we obtain a similar space which is Fréchet-Urysohn under the assumption that $\mathfrak p = \mathfrak b$ or $\mathfrak b = \mathfrak c$.

We will survey several results, obtained with various coauthors, showing the impact of infinite games on the theory of cardinal functions in topology.

In a recent paper, Cummings, Eisworth and Moore answered two questions of Baumgartner by proving that the diamond principle ◊(omega_1) is sufficient to construct a minimal non-σ-scattered linear order which is an Aronszajn line and not a Souslin line. They also gave the first consistent example of a minimal non-sigma-scattered linear order of size greater than Aleph_1. Their construction works for arbitrary successor cardinals, and leaves open the case of inaccessibles. In this talk, we will give a construction that covers this missing case.

In this work, joint with Iian Smythe, we describe a unified descriptive set theoretic framework for the study and comparison of classification problems for various classes of manifolds. Within this framework, we record several fundamental results, on the Borel complexity of the homeomorphism problem for compact manifolds or 2-manifolds, for example, and of the isometry problem for hyperbolic manifolds, and for algebraically finite hyperbolic manifolds of low dimension. We will close with a list of some of the most conspicuous open questions.

Using the Whitehead problem as a motivating example, we will discuss a number of emerging connections between set theory and condensed mathematics, a newly developed framework for applying algebraic methods to the study of objects carrying topological structures. In particular, we will highlight the central role played by extremally disconnected compact Hausdorff spaces in the theory and the concomitant connections with set theoretic forcing. The talk will contain joint work with Jeffrey Bergfalk and Jan Šaroch.

The topics of this introductory talk are construction schemes and capturing axioms. Firstly introduced by Stevo Todorčević, construction schemes have worked as a tool for building uncountable objects by means of finite approximations. Capturing axioms, which hold in many canonical models of ZFC, asert the existence of construction schemes with strong combinatorial properties that can be used to deduce the existence of several objects which are known to be indepentent from the usual Axioms of Set Theory. We will be discussing the state of art regarding construction schemes, and will also present some open problems on this subject. Joint work with Osvaldo Guzmán and Stevo Todorčević.

(joint work with Tom Benhamou) The Tukey structure of ultrafilters on $\omega$ has been studied extensively in the last two decades with various works of Blass, Dobrinen, Kuzeljevic, Mijares, Milovich, Raghavan, Shelah, Todorcevic, Trujillo, and Verner. Research on the Galvin property for ultrafilters over uncountable cardinals, in particular on measurable cardinals, has gained recent momentum, due to applications in infinite combinatorics, cardinal arithmetic, and inner models and forcing theory, with various works of Benhamou, Garti, Gitik, Poveda, and Shelah. Joint work with Tom Benhamou began with the observation that the Galvin property is equivalent to being not Tukey maximal; hence, Tukey types refine various Galvin properties. We initiate the development of the Tukey theory of ultrafilters on measurable cardinals, allowing the flow of results from the countable to the uncountable and vice versa. The situation for ultrafilters on measurable cardinals turns out to be quite different from that on $\om$, sometimes greatly simplifying the situation on $\omega$ and sometimes posing new obstacles. The structure of the Tukey classes also turns out to be sensitive to different large cardinal hypotheses. We will present results from our preprint arXiv:2304.07214 and ongoing work.

Raghavan and Todorcevic recently proved that the existence of a Woodin cardinal imply an old conjecture of Galvin holds: if we color the pairs from an uncountable set of reals, then we can find a set homeomorphic to the rationals on which the coloring assumes at most two values. We show that it suffices to assume the existence of a Ramsey cardinal, and that Galvin's conjecture does not imply 0-sharp exists.

In recent years we have witnessed an unprecedented confluence of methods from discrete and continuous mathematics, especially in subjects having to do with logic and topology. One can cite fields such as continuous model theory, homotopy type theory and, most relevant to this talk, combinatorial limits. The latter have started from the notion of graphons and have been generalised to other objects, including the very general Stone pairings. In this subject one looks at uncountable limits of a countable sequence of finite objects, with various logical properties that carry through. In the context of first order logic, one can think of Los’s theorem for ultraproducts, but various other transfer theorems have been obtained in this other contexts. In this talk we shall review some of these notions, including a Ramsey theorem about ultraproducts, and then connect them with the study of abstract logics through new satisfaction relations.

Given a relation R on the Baire space ω^ω like the eventually dominating ordering $≤^∗$ and an ideal I on ω^ω like the meager ideal M or the null ideal N , Switzer defined natural bounding numbers $\mathfrak b(R_I)$ and dominating numbers $\mathfrak d(R_I)$. While these two cardinals are dual under CH, this duality breaks down in models obtained by adding many reals. Namely, with one exception, the b(RI) can be consistently ℵ1 while the d(RI) are always at least $\mathfrak c^+$. Furthermore, the latter do not depend on the ideal while the former can be consistently distinct for different ideals. Moreover, some of the bounding numbers can be computed from the values of cardinal invariants in Cicho\'n’s diagram while others cannot. We shall also see that one can restore duality by introducing two more natural cardinals associated with R and I. In my talk, I will first discuss older results from joint work with Switzer and then present recent work (still in progress).

In 2010, given $A\subseteq \Bbb R$, Hattori defined the topology $\tau(A)$ on $ \Bbb R$ making a "combination" of the Sorgenfrey topology and the Euclidean topology. The topological space $( \Bbb R,\tau(A))$ is denoted by $H(A)$ and it is called the Hattori space associated to $A$. In this talk, we will define the notion of almost topological group and its reflection group. In particular, the Sorgenfrey line is an almost topological group and $ \Bbb R$ is its reflection group. Then we extend the Hattori's definition to almost topological groups. For a subset $A$ of an almost topological group $G$, we define the Hattori space $H(A, G)$, where $H(A, G)$ is a topological space whose underlying set is $G$ and whose topology is defined as follows: if $x\in A$ (respectively, $x\notin A$), then the neighborhoods of $x$ in $H(A, G)$ are the same neighborhoods of $x$ in the reflection group (respectively, $G$). The main goal of this talk is to study some topological properties in the space $H(A, G)$. For instance, second-countability, network weight, metrizability, local compactness, etc.

We shall discuss how to consistently distinguish various properties between the Fréchet-Urysohn and Pytkeev ones in the realm of $C_p$-spaces over sets of reals. Instead of working with these local properties of spaces of functions directly, we deal with corresponding combinatorial covering properties of the domain spaces, thus using known characterizations proved by various authors in the framework of so-called local-global duality in $C_p$-theory. The central role plays the introduction of ideals on natural numbers as parameters and using the Kat\v{e}tov preorder (and variations thereof) on them. The talk will be based on a recent work with S. Bardyla and J. \v{S}upina.

All spaces are assumed to be separable and metrizable. Ostrovsky showed that every zero-dimensional Borel space is σ-homogeneous. In this talk, we will discuss the following results, which were inspired by Ostrovsky's theorem: (1) Assuming AD, every zero-dimensional space is σ-homogeneous, (2) Assuming AC, there exists a zero-dimensional space that is not σ-homogeneous, (3) Assuming V=L, there exists a coanalytic zero-dimensional space that is not σ-homogeneous. Along the way, we will introduce two notions of hereditary rigidity, and give alternative proofs of results of van Engelen, Miller and Steel. It is an open problem whether every analytic zero-dimensional space is σ-homogeneous. This is joint work with Zoltán Vidnyánszky.

The purpose of this session is to generate research questions and project ideas for students, including undergraduates. The first half will be a discussion of the nuts-and-bolts of leading student research, especially students who have little background in topology or set-theory. We will see examples of successful projects of different levels and durations. The second half will be a problem session where participants share ideas for student projects. The workshop organizers will collect and distribute the ideas collected from the session. Participants are encouraged to bring problems and/or student research strategies to share.