Talks involving set-theoretic and other techniques used to investigate topics in general topology. Organizers: Will Brian and Steven Clontz
We will present some results on existence of dense functionally countable subspaces in spaces $C_p(X)$. It will be shown, among other things, that there is a consistent example of a scattered Lindel\"of $P$-space $X$ for which $C_p(X)$ has no dense functionally countable subspace and that $\mathbb R^{\omega_1}$ has a dense functionally countable subspace of cardinality $\omega_2$ if and only if the Kurepa Hypothesis holds.
View Submission
We discuss a natural topology on powers of a space that is inspired by the Vietoris topology on compact subsets. We then place this topology in context with other product topologies; specifically, we compare this topology with the Tychonoff product, the box product, and Bell's uniform box topology. We identify a variety of topological properties for the specific case when the ground space is discrete. When the ground space is the Euclidean real line, we show that the resulting power is not Lindelöf, and hence, not Menger. This shows that, unlike the the Vietoris topology on unordered compact subsets, covering properties of the ground space need not transfer to the Vietoris power.
View Submission
We present an regular space that is not completely regular but only barely so: not only is it first-countable, but in addition all closed sets are $G_\delta$-sets and all points are zero-sets. This answers a question about how the lattice of zero-sets is situated in the lattice of all open sets. Some intermediate results on the Niemytzki plane make excellent homework exercises for a topology course.
View Submission
The concept of a C-space was introduced in 1978 by D. Addis and J. Gresham in order to provide a new class of spaces in dimension theory. We present an internal characterization for an inverse system $\mathbf{X}$ of compact Hausdorff spaces and maps that shows when its limit will be a C-space. This is precisely when $\mathbf{X}$ is a ``C-system,'' whose definition will be given in this presentation. We use this characterization to construct a C-system $\mathbf{Y}$ so that its inverse limit is a weakly infinite-dimensional, strongly countable-dimensional, metrizable compactum that is a C-space. Finally we introduce a new notion into topological game theory called a game-theoretic C-system.
View Submission
Combinatorial covering properties as Rothberger’s, Hurewicz’s and Menger’s are procedures for generating a cover of a given topological space from a sequence of covers of this space. We present the most celebrated such properties together with the most important examples in a classical countable case. We also explore how these notions and examples extend to the uncountable context, where the initial sequence of covers has length $\kappa$ for some uncountable cardinal $\kappa$. In this generalized setting, we replace the Cantor space $2^\omega$ and the classical Baire space $\omega^\omega$ with the $\kappa$-Cantor space $2^\kappa$ and the $\kappa$-Baire space $\kappa^\kappa$, respectively. This is joint work with Piotr Szewczak and Lyubomyr Zdomskyy.
View Submission
The *Cantor's Fan* is a planar topological space in $\mathbb{R}^2$, constructed from the Cantor set in $[0,1]$ and inspired by the *Cantor's Teepee* introduced in 1921 by Bronisław Knaster and Kazimierz Kuratowski. In this paper, we determine which properties of the Cantor's Teepee persist in the Cantor's Fan; we restate the main properties in contemporary language, provide complete formal proofs, and include illustrative figures.
View Submission
We will discuss joint work with Juris Steprāns concerning the countable dense homogeneity of products of Polish spaces, with a focus on uncountable products. Our main result states that a product of fewer than $\mathfrak{p}$ Polish spaces is countable dense homogeneous if the following conditions hold: (1) Each factor is strongly locally homogeneous, (2) Each factor is strongly $n$-homogeneous for every $n\in\omega$, (3) Every countable subset of the product can be brought in general position. For example, using the above theorem, one can show that $2^\kappa$, $\omega^\kappa$, $\mathbb{R}^\kappa$ and $[0,1]^\kappa$ are countable dense homogeneous for every infinite $\kappa<\mathfrak{p}$ (these results are due to Steprāns and Zhou, except for the one concerning $\omega^\kappa$). In fact, as a new application, we showed that every product of fewer than $\mathfrak{p}$ connected manifolds with boundary is countable dense homogeneous, provided that none or infinitely many of the boundaries are non-empty. This generalizes a result of Yang.
View Submission
The proximal game, introduced in 2014, is a two-player infinite game played in a uniform space. It relies on the uniform structure in an inherent way: the first player chooses elements of the uniformity while the other selects points. A winning strategy for the first player implies the space has certain additional topological properties, which as such are independent of the particular uniform structure with which the game was played. So, is the uniform structure really necessary? I will discuss some recent progress in divorcing the proximal game from its reliance on a uniform structure, resulting in the creation of purely topological "point-star" games.
View Submission
In 1996, Stares introduced monotonically semi-neighborhood refining (MSNR) spaces and showed that well-ordered neighborhood (F) spaces are MSNR spaces. In this talk, the question of whether MSNR spaces have well-ordered neighborhood (F) is explored. We show that MSNR spaces have well-ordered (F) and hence are monotonically normal and hereditarily paracompact. MSNR spaces are also shown to be lob-spaces. The relationships between MSNR spaces with other monotone covering properties will also be explored.
View Submission
Session participants are invited to join an interactive problem session discussing open questions in our discpline.
View Submission
Given an infinite cardinal $\kappa$ and a $\kappa$-splitting $\kappa^+$-tree $T$, we topologize $T$ as follows: if $x\in T$, then sets of the form $\uparrow x \setminus \uparrow F$, for $F$ a set of immediate successors of $x$ with $|F|<\kappa$, form a basis of neighbourhoods of $x$. We then ask whether $T$ is $\kappa^+$-compact with respect to this topology and characterize this property in purely order-theoretic terms. Such trees are necessarily $\kappa^+$-Aronszajn, so they may (consistently) not exist when $\kappa\ge\aleph_1$. In this talk, discuss how to construct such trees using Proxy Principles, introduced by Brodsky and Rinot. We will also mention a further consitency result on the non-existence of such trees together with the failure of the tree property at $\aleph_2$. This is joint work with Ari Meir Brodksy.
View Submission
The notion of hyperconvexity of metric spaces was introduced by Aronszajn and Panitchpakdi in 1956 in their study of extensions of uniformly continuous mappings between metric spaces. From the topological point of view, a hyperconvex space is an absolute retract via a nonexpansive retraction. By the theorem of Nachbin and Kelley, hyperconvex real Banach spaces can be treated as Stonian spaces C(K) of all real-valued continuous functions on extremally disconnected compact Hausdorff spaces K. On the other hand, the notion of a partial metric space was introduced by Matthews in 1994. He showed, roughly speaking, how metric-like tools can be extended to non-Hausdorff topologies, and he also indicated some applications of this class of spaces in the study of the denotational semantics of programming languages. Further applications of partial metrics can also be found in the geometry of Banach spaces. In this talk, we present several different approaches to defining hyperconvexity in partial metric spaces. In particular, we show that the analogue of the Aronszajn–Panitchpakdi notion of hyperconvexity fails to possess certain key properties present in the classical metric setting. Finally, we outline some perspectives for further research.
View Submission
The Cofinality of Generating Familes (joint with Paul Gartside) Cofinal sets, which study what happens "eventually" in a given order, are ubiquitous in mathematics. In Topology, they play a particularly noticeable role. Consider, for example, a subspace $M$ of the reals. The topology can be captured by convergent sequences, and hence we can ask the question: how many sequences are needed to characterize the topology on $M$? As another example, consider questions (essentially about cofinalities) such as: how many compact sets cover $M$ (the "compact covering number" of $M$)? or how many compact subsets generate the topology on $M$ (the $k$-ness number of $M$)? In this talk, we will discuss recent work of Paul Gartside and the speaker on these questions. We will show how these questions about how many sequences (or compact sets) generate the topology of $M$ can be viewed through the lens of the Tukey order on relations and how, as such, we can apply set-theoretic techniques. We show how certain cardinal invariants are used to answer these topological questions. Finally, we discuss how (in light of recent work of James Cummings and the speaker on extender-based forcing and mod-finite scales) we can, among spaces with a fixed value of the compact covering number, the $k$-ness number can be quite varied.
View Submission
The [$\pi$-Base community database of topological counterexamples](https://topology.pi-base.org/) was recognized in Fall 2025 as the highest-voted [crowdsourced math project on Terence Tao's MathOverflow list](https://mathoverflow.net/questions/500720/list-of-crowdsourced-math-projects-actively-seeking-participants). While much can be done by simply modeling Objects/Spaces, Properties, and Theorems, without a notion of set theory and cardinality, we quickly find limitations, for example: - Several "open questions" on $\pi$-Base are equivalent to the Continuum Hypothesis - Thirteen properties on $\pi$-Base are just different cardinalities, with explicit theorems written to connect them. We will discuss the current plan to incorporate results from set theory into the $\pi$-Base, and seek input from potential users.
View Submission
The $\pi$-Base is a database devoted to general topology, listing topological spaces, their properties, and theorems relating properties together. In this talk, I will show how I used the $\pi$-Base as an undergraduate student to discover interesting research problems. As an example, I will show how the $\pi$-Base led to me proving that a non-semiregular almost discrete space must be the disjoint union of the Sierpiński space and a discrete space.
View Submission