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General and Set-Theoretic Topology

Submissions closed on 2025-06-15 11:59PM [Central Time (US & Canada)].

Talks involving set-theoretic and other techniques used to investigate topics in general topology. Organizers: Chris Caruvana, Jared Holshouser, Lynne Yengulalp.

Set-theoretic topology is the study of techniques from set theory that are used to investigate topics in general topology. Such topics include topological games, homogeneity, hyperspaces, topological algebra, compactifications, and much more. Problems about topological topics or properties often require extra axioms of set theory to answer. Advances in set theory, conversely, have applications in general topology. Recent work by speakers in this session include set-theoretic schemes that build uncountable structures from finite ones (Osvaldo Guzman), persistent properties in uniform box products (Jocelyn Bell), resolvability in certain classes of topological spaces (Lajos Soukup), compact generators in function spaces (Paul Gartside), cardinal inequalities and pi-character (Ivan Gotchev and Vladimir Tkachuk), zero-dimensional sigma-homogeneous spaces (Andrea Medini), and others.

Organizers

Accepted Submissions:

A measure of Isbell-convexity for a quasi-metric space — Collins Amburo Agyingi

Let (X,d) be a T0-quasi-metric space. Then it has been shown that X has a q-hyperconvex hull which is denoted by QX. It is known that every q-hyperconvex T0-quasi-metric space is bicomplete. However, the converse is not true, that is, there exist bicomplete T0-quasi-metric spaces that are not q-hyperconvex. In this talk, we shall present a parameter that measures how far a bicomplete T0-quasi-metric space is from being hyperconvex. We will present some characteristics of this new parameter.

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An iterable forcing property and universally meager sets — Valentin Haberl

By a space we mean a metrizable separable zero-dimensional space. A space X⊆2ω is universally meager if for any Polish space Y and any continuous nowhere constant map f:Y→2ω the preimage f−1[X] is meager in Y. We call a space totally imperfect if it contains no copy of 2ω. We present a forcing property (†), which is a strenthening of properness and implies that no dominating reals are added. It is known that many classical forcing posets like Cohen, Sacks and Miller satisfy this property. We showed that property (†) is preserved by countable support iterations. We then used this preservation result to prove that if we have such an iteration of length ω2 over a model of CH, where the single forcings have size at most ω1, all universally meager sets X⊆2ω have size at most ω1 in the forcing extension. \ This has multiple set-theoretic applications: In the Miller model, we generalized our result of having no concentrated and γ-sets of size continuum to totally imperfect Hurewicz sets, which are universally meager by a result of Zakrzewski. Moreover, since Bartoszyński showed that all perfectly meager spaces are universally meager in the Miller model, we get that indeed even all perfectly meager spaces have size stricly less than continuum in the Miller model. Miller proved in 2005 that there exists a strong measure zero set of size ω1 iff there exists a Rothberger space of size ω1. Goldstern, Judah and Shelah constructed in 1993 a forcing iteration for which there is a strong measure zero set of size ω2 in the extension. However, this iteration satisfies property (†) and Rothberger spaces are universally meager in this model. Hence our result implies that it is consistent with ZFC to have a strong measure zero set of size ω2, but no Rothberger spaces of size ω2. This is joint work with Piotr Szewczak (Cardinal Stefan Wyszyński University in Warsaw) and Lyubomyr Zdomskyy (TU Vienna).

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Cardinal bounds in spaces with a π-base whose elements have an H-closed closure — Davide Giacopello

We deal with the class of Hausdorff spaces having a π-base whose elements have an H-closed closure. Carlson proved that |X|≤2wL(X)ψc(X)t(X) for every quasiregular space X with a π-base whose elements have an H-closed closure. We provide an example of a space X having a π-base whose elements have an H-closed closure which is not quasiregular (neither Urysohn) such that |X|>2wL(X)χ(X) (hence, |X|>2wL(X)ψc(X)t(X)). Still in the class of spaces with a π-base whose elements have an H-closed closure, we establish the bound |X|≤2wL(X)k(X) for Urysohn spaces and we give an example of an Urysohn space Z such that k(Z)<χ(Z). Lastly, we present some equivalent conditions to the Martin's Axiom involving spaces with a π-base whose elements have an H-closed closure and, additionally, we prove that if a quasiregular space has a π-base whose elements have an H-closed closure then such a space is Baire.

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Combinatorial covering properties in countable and uncountable contexts — Michał Pawlikowski

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 κ for some uncountable cardinal κ. In this generalized setting, we replace the classical Baire space ωω with the generalized Baire space κκ. This is joint work with Piotr Szewczak and Lyubomyr Zdomskyy.

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Combinatorial structures of concentrated sets — Piotr Szewczak

Let X be a set of reals and κ be an uncountable cardinal number. The set X is κ-concentrated, if X has size at least κ and contains a countable set D such that each closed subset of X, disjoint with D, has size smaller than κ. Various forms of concentrated sets play an important role in the study of combinatorial covering properties such as Rothberger’s, Hurewicz’s, and Menger’s properties. We investigate the behavior of such sets in different models of set theory. This is a joint work with Michał Pawlikowski and Lyubomyr Zdomskyy. The research was funded by the Polish National Science Center and Austrian Science Fund; Grant: Weave-UNISONO, Project: Set-theoretic aspects of topological selections 2021/03/Y/ST1/00122.

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Countable dense homogeneity and topological groups — Andrea Medini

All spaces are assumed to be separable and metrizable. A space X is countable dense homogeneous (CDH) if all countable dense subsets of X can be mapped onto each other by homeomorphisms of X. The fundamental theorem of countable dense homogeneity states that every "sufficiently homogeneous" Polish space is CDH. This result motivated a long-standing search for examples of non-Polish CDH spaces. We contribute to this line of research by exhibiting a non-Polish CDH topological group. This is joint work with Claudio Agostini and Lyubomyr Zdomskyy.

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Frechet spaces and M-separable (selectively separable) spaces — Alan Dow

Frechet spaces are selectively separable. Finite products of countable Frechet spaces need not be Frechet but it is independent as to whether they are selectively separable. We will review some recent results on this topic and, at the moment, think we have a new one.

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Holey Vietoris--Rips complex, Batman! — Chris Wells

Given a metric space and a positive number d, the Vietoris--Rips (VR) complex of scale d is the simplicial complex whose faces are all sets of diameter at most d. Recently, there's been a push to understand the VR * How many holes (non-trivial homologies) are there? * How big is the largest facet? * How small is the smallest facet? * How many differently-sized facets are there? Based on joint work with Joe Briggs and Ziqin Feng.

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In the quest for squares in plane continua — Cristina Villanueva-Segovia

In this talk we will be looking at conditions on a plane continuum X (not necessarily locally connected) that guarantee the existence of four points in X that are the vertices of a Euclidean Square (in which case we say that X admits an inscribed square). In particular we show that ''certain type of square inscription´´ is generic among continua that separate the plane. The motivation of this work comes from the square peg problem: Does every Jordan curve admits an inscribed square?

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Metrization theorem in forcing extensions — Akira Iwasa

We study ``metrizaion theorem in forcing extensions.'' That is, for a non-metrizable space X, we study what topological property X has to have to become metrizable in forcing extensions. We provide such property for a class of spaces with weight ≤κ and each point has a neighborhood of density <κ, where κ is a regular uncountable cardinal.

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On the modular metric topology — Olivier Olela-Otafudu

Following earlier authors in this subject, the topology induced by a modular metric is herein called a modular topology. We show that such a topology is metrizable. More precisely, we show that the uniform topology induced by the uniformity on the modular set of a modular pseudometric is metrizable. In addition, we observe that such a topology is coarser than the underlying topology of the uniformity induced by the corresponding pseudometric. Other related immediate observations are also presented. Unknown environment 'references'

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Partial metric spaces - topological properties and applications — Dariusz Bugajewski

The notion of a partial metric space was introduced by Matthews in 1994 who showed, roughly speaking, how metric--like tools can be extended to non--Hausdorff topologies. He also indicated some applications of this class of spaces in the study of denotational semantics of a programming language. In this talk we are going to present some necessary and sufficient conditions under which the topology generated by a partial metric is equivalent to the topology generated by a suitably defined metric. Next, we are going to focus on two basic topological properties of partial metric spaces, namely completeness and compactness. In particular, it appears that in these spaces compactness is equivalent to sequential compactness. Finally, we will focus on a very general fixed point theorem for mappings acting in partial metric spaces. In that theorem we impose some conditions on behavior of considered mappings on orbits and a condition relating orbits of points of small size.

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Pseudo-ℵ1-compactness in R-factorizable groups — Olga Sipacheva

This is a joint work with Evgenii Reznichenko. A topological group is said to be R-factorizable if, given any continuous function f:G→R, there exists a continuous homomorphism h:G→H to a second-countable topological group H and a continuous function g:H→R such that f=g∘h. The main unsolved problems of the theory of R-factorizable groups are as follows: 1. Is the property of being an R-factorizable group topological? In other words, is any topological group homeomorphic to an R-factorizable one R-factorizable? 2. Is the square of an R-factorizable group R-factorizable? 3. Is any R-factorizable group pseudo-ℵ1-compact, that is, contains no uncountable locally finite family of open sets? 4. Is the image of an R-factorizable group under a continuous homomorphism R-factorizable? We show that if the answer to question 2 is positive, then so is the answer to question 1. Also, if the answer to question 4 is positive, then so is the answer to question 3, and if the answer to question 3 is negative, then so are the answers to questions 1 and 2. Note that there are examples of R-factorizable groups G and H such that G×H is not R-factorizable. Our main concern is the pseudo-ℵ1-compactness of R-factorizable groups. We prove that an R-factorizable group G is pseudo-ℵ1-compact if it satisfies any of the following conditions: (i) the weight of G is at most ω1; (ii) the pseudocharacter of G equals ω1; (iii) G2 is R-factorizable; (iv) G contains a nonmetrizable compact subspace; (v) G contains a Lindel\"of subspace of uncountable pseudocharacter.

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Q-SETS, ∆-SETS AND L-SPACES — Pourya Memarpanahi

The concept of a ∆-set of reals was originally defined by G.M. Reed. An equivalent version was defined by Eric van Douwe and later on was generalized to an arbitrary topology space, (∆-space) . Historically, this notion arose in the study of the normal Moore space conjecture, where Q-sets were used to construct important counterexamples to the conjecture. We prove that Moore's L-space (a hereditarily Lindelöf but not separable space in ZFC) is not a Q-set space and if Aronszajn tree naturally associated with Moore’s L-space is special Moore's L-space will not be a ∆-space.

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Reflecting topological properties in closures of countable sets. — Vladimir Tkachuk

This talk's purpose is to present some results on whether a topological space X has a property P given that A― has P for any countable set A⊂X. The respective line of research was outlined in a recent paper of A. Dow and the author. We prove, among other things, that there is a consistent example of a metric space X such that A― is \v Cech-complete for any countable A⊂X but X has no dense \v Cech-complete subspace. This talk will also feature some related results on general locally convex spaces.

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Sequential and countable compactness associated to Ramsey-like properties — Cesar Corral

In this talk, we will examine variants of sequential compactness and countable compactness that are associated with Ramsey-like properties. These notions have arisen naturally in various topological and combinatorial contexts. We will present several results obtained by treating these compactness properties as central objects of study. The talk will conclude with some connections to classical problems and open questions.

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Stone-Cech extensions of pseudocompact convex spaces. — Evgenii Reznichenko

All spaces are assumed to be Tychonoff spaces. Let X be a convex pseudocompact subspace of some locally convex space (LCS). Question 1. Is it true that the Stone-Cech extension βX has the structure of a convex compact set? Is it true that βX is homeomorphic to a convex compact subset of some LCS? The answer to this question is positive if X=P(Y), where P(Y) is the space of probability Radon measures on X in the weak topology [1]. In this case, Y is a pseudocompact space and βP(Y)=P(βY). There is a convex compact set K and its dense convex pseudocompact subset C such that βC≠K. Proposition 1. βX is path-connected. This fact is related to the fact that the structure of a convex set with X extends to βX. A space S with a (separately) continuous operation p:[0,1]×X×X→X is called a (semi)topological convex set if there is an embedding of S into a linear space (without topology) such that p(λ,x,y)=λx+(1−λ)y. Theorem 1. If S is a pseudocompact topological convex set, then βS is a semitopological convex set. Clearly, a convex subset of some LCS is a topological convex set. Theorem 1 implies Proposition 1. Theorem 2. If S is a topological convex set and S2 is pseudocompact, then βS is a topological convex set. Theorem 3. If S is a countable compact semitopological convex set, then βS is a semitopological convex set. The theorems imply that the convex set structure from S extends to βS. A (semi)topological convex set S is a universal (semi)topological algebra with continuum operations pλ:S×S→S, pλ(x,y)=p(λ,x,y), where λ∈[0,1]. The signature of S is continuous, is a segment of [0,1]. The theorems are proved using results on the extension of operations in universal algebras obtained in [2]. [1] Reznichenko, E., "Stone-Cech extensions of probability measure spaces." arXiv preprint arXiv:2412.11838 (2024). [2] Reznichenko, E., "Extensions and factorizations of topological and semitopological universal algebras." Topology and its Applications (2025): 109256.

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The Borel hierarchies of function spaces consisting of metrics — Katsuhisa Koshino

Function spaces have been studied in the theory of infinite-dimensional topology, and their Borel hierarchies play important roles in recognizing topologies on them. In this talk, we shall investigate the Borel hierarchies and the complete metrizability of function spaces consisting of metrics on metrizable spaces, and as an application, we will decide their topological types.

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The F_sigma-below relation and some new separation axioms in the category of locales — Mbekezeli Nxumalo

We use F_sigma sublocales to define a relation on locales called the F_sigma-below relation. This relation is weaker than the rather below relation. Two separation axioms, namely weakly D-completely regularity and F_sigma-regularity, between perfectness and weakly subfitness are introduced using the F_sigma-below relation. We discuss properties of these two locales and find their relationship with other locales such as regularity and subfitness.

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Topological spaces after forcing — Pedro Marun

If (X,τ) is a topological space and P is a poset, then τ may cease to be a topology after forcing with P, for example if new subsets of X are added. Nevertheless, in the generic extension, τ is a basis for a topology, call it τP, which is finer than τ. One can then ask which properties of τ are inherited by τP. In this talk, we will look at what happens to the Lindelöf property under different classes of forcing notions.

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