Submissions (41)
Accepted (37):
$\mathbb H^*$ — Will Brian <wbrian.math@gmail.com>
Let $\mathbb H$ denote the half-line $[0,\infty)$, and let $\mathbb H^* = \beta \mathbb H \setminus \mathbb H$ denote its \v{C}ech-Stone remainder. We aim to discuss a recent theorem showing that the Continuum Hypothesis ($\mathsf{CH}$) implies $\mathbb H^*$ is the ``generic'' continuum of weight $\aleph_1$. What precisely this means is the main topic of the talk, but roughly it means that, in the appropriate generalized sense, $\mathsf{CH}$ implies $\mathbb H^*$ is the inverse Fra\"{i}ss\'e limit of the class of metrizable continua. This leads directly to a topological characterization of $\mathbb H^*$ under $\mathsf{CH}$: i.e., a topological property of $\mathbb H^*$ such that $\mathsf{CH}$ implies $\mathbb H^*$ is, up to homeomorphism, the only weight-$\aleph_1$ continuum with this property.
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A quantitative measure of non-Isbell convexity in T_0-quasi-metric spaces — Collins Amburo Agyingi <agyingic@gmail.com>
Isbell-convex (q-hyperconvex) T<sub>0</sub>-quasi-metric spaces are known to be bicomplete, although the converse does not hold in general. This talk introduces a quantitative invariant that measures the extent to which a bicomplete T<sub>0</sub>-quasi-metric space fails to be Isbell-convex. The invariant is defined as a two-component parameter h<sub>q</sub>(X)=((h<sub>q</sub>)<sub>1</sub>(X),(h<sub>q</sub>)<sub>2</sub>(X)) capturing the inherently asymmetric forward and backward deviations from convexity. Using the framework of minimal function pairs arising in the Isbell hull, we show that h<sub>q</sub>(X)=(0,0) characterizes Isbell-convexity. We further establish key properties of this invariant, including non-negativity, invariance under isometries, and monotonicity with respect to subspaces and nonexpansive maps, as well as boundedness in the finite case. Concrete examples demonstrate how bicomplete spaces may fail to be Isbell-convex and illustrate how the invariant quantifies this deviation. This provides a new quantitative tool for studying the interplay between convexity and completeness in asymmetric topology.
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A soft version of Magill's theorem on n-point Hausdorff compactifications — Robert Gryczka <robert.jakub.gryczka@gmail.com>
Magill's theorem on necessary and sufficent conditions for topological space to have n point Hausdorff compactification is one of the classical tools in the theory of compactifications of topological spaces. In this talk, a soft version of this theorem will be presented in the context of soft topological spaces. Basic concepts of soft set theory and soft topology will be discussed, with particular emphasis on soft Hausdorff compactifications and their relationships with classical topological constructions. Subsequently, a generalization of Magill's theorem to the framework of soft topology will be presented, together with conditions characterizing the existence of soft n-point Hausdorff compactifications.
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A topological Ramsey space for pseudotrees — David Chodounsky <chodounsky@math.cas.cz>
Topological Ramsey spaces introduced by Todorcevic are spaces equipped with further structure (an ordering, a notion of approximation) which satisfy certain axioms. These spaces can be seen as abstract variations of the classical Ellentuck space and provide a unified framework for proving combinatorial partition theorems. We introduce a new type of A topological Ramsey space consisting of certain strong (sub)trees, which can be can be used for coding rational countable pseudotrees.
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A tree-based perfectly normal space whose square is not countably metacompact — Assaf Rinot <rinotas@math.biu.ac.il>
The study of the interval topology of trees traces back to the 1960's with Jones' work on the normal Moore space problem. There are various limitations on the kind of spaces that can be obtained this way, for instance, Nyikos proved that for every tree $T$, if $X_T$ is normal, then it is also countably metacompact (CMC), i.e., there are no Dowker trees. Throughout the years, many consistency results were proven concerning the topological characteristics of spaces of the form $X_T$, but we couldn't find similar results dealing with the spaces' square. Here, we present a consistent construction of an Aronszajn tree $T$ such that $X_T$ is perfectly normal but $(X_T)^2$ it not even CMC. A key component of the construction is the use of `elevators' -- a device that enables to construct the tree level-by-level while optimally controlling features of its powers.
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Analogues of Hindman's Theorem for Topological Groups — Serhii Bardyla <sbardyla@gmail.com>
We shall discuss the partition regular properties of topological groups, and obtain an extension of Hindman’s theorem where the monochromatic sets of finite sums are required to satisfy additional topological constraints. In particular, our results imply that for every nowhere dense subset $C\subseteq \mathbb R^n$ there exists an open set $P\supseteq C$ such that for every finite coloring of $\mathbb Q^n\setminus P$ there exists a family $\mathcal A$ of sequences in $\mathbb Q^n\setminus P$ satisfying the following conditions: (i) the set of finite sums $\operatorname{FS}(A)$ is a closed discrete subset of $\mathbb R^n$ for all $A\in\mathcal A$; (ii) the set $\bigcup_{A\in\mathcal A}\operatorname{FS}(A)$ is monochromatic; and (iii) the set $\bigcup_{A\in\mathcal A}\operatorname{FS}(A)$ is dense in an open unbounded subset of $\mathbb R^n$. The aforementioned result was obtained using a new characterization of spaces whose Stone-Čech compactifications possess remote points.
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- Plenary
Classification of countably tight groups. — Alex Shibakov <ashibakov@tntech.edu>
We provide a classification of convergence structures in countably tight groups using the Invariant Ideal Axiom and the related Definable Ideal Axiom. We then show some applications to countable groups and the boolean groups with the bounded topology, answering a few published questions. A number of open problems will also be mentioned.
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Concentrated sets and the Hurewicz property — Valentin Haberl <valentin.haberl.math@gmail.com>
A set of reals $X$ is $\mathfrak{b}$-concentrated if it has cardinality at least $\mathfrak{b}$ and it contains a countable set $D\subseteq X$ such that each closed subset of $X$ disjoint from $D$ has size smaller than $\mathfrak{b}$. $\kappa$-concentrated sets play a crucial role in the investigation of combinatorial covering properties. It is independent of ZFC if all $\mathfrak{b}$-concentrated sets are Hurewicz. We present ZFC results about how structures of $\mathfrak{b}$-concentrated sets with the Hurewicz covering property can be characterized with the notion of being meager-unbounded, which we then use to get that such structures are productively Hurewicz. We obtain that assuming that the semifilter trichotomy holds each $\mathfrak{b}$-concentrated set is Hurewicz and even productively Hurewicz. In particular, in the Miller model $\mathfrak{b}$-concentrated sets are also productively Rothberger. We analyze the Laver model, where the behavior of Hurewicz $\mathfrak{b}$-concentrated sets differs from the one under the semifilter trichotomy. This is joint work with Piotr Szewczak (University of Warsaw) and Lyubomyr Zdomskyy (TU Vienna).
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Constructions of crowded zero-dimensional Hausdorff P-spaces without the Axiom of Choice — Eliza Wajch <eliza.wajch@gmail.com>
#Constructions of crowded zero-dimensional Hausdorff _P_-spaces without the Axiom of Choice **Eliza Wajch** ***Institute of Mathematics, University of Siedlce, 3 Maja 54, 08-110 Siedlce, Poland*** The results presented here form part of the author's joint work with Eleftherios Tachtsis [2]. They are motivated by the question posed in [1]: is the existence of a non-discrete Tychonoff _P_-space provable in **ZF**? A topological space whose every _G_<sub>δ</sub>-set is open is called a _P_-space. Throughout, our set-theoretic framework is **ZF** or **ZFA**. Among several related results, we show that non-empty zero-dimensional crowded Hausdorff _P_-spaces exist in every permutation model of **ZFA** and in every model of **ZF** having an aleph of uncountable cofinality. A key ingredient in our analysis is the following construction of zero-dimensional Hausdorff spaces which, under additional hypotheses, yields _P_-spaces. Let _X_ be an infinite set, and let 𝒵 be a family of subsets of _X_ closed under finite unions and containing [_X_]<sup><*ω*</sup>. For _x_ ∈ [X]<sup><*ω*</sup> and _z_ ∈ 𝒵 with _x_ ∩ _z_ = ∅, define _B_(_x_,_z_) = \{_y_ ∈ [_X_]<sup><ω</sup> : _x_ ⊆ _y_ ⊆ _X_ \ _z_}. Let 𝒯 be the topology on [_X_]<sup><*ω*</sup> such that, for each _x_ ∈ [_X_]<sup><*ω*</sup>, the family {_B_(_x_,_z_): _z_∈ 𝒵 and _x_ ∩ _z_ = ∅} forms a neighborhood base at _x_. The resulting space **S**(_X_, 𝒵) = ([_X_]<sup><*ω*</sup> , 𝒯) is Hausdorff and zero-dimensional, and it is crowded whenever _X_ is not a member of 𝒵. Assuming that 𝒵 is a bornology on _X_, we obtain that the space **S**(_X_, 𝒵) is homogeneous, and if it is a _P_-space, then 𝒵 is a σ-ideal. In particular, **S**(_X_, [_X_]<sup><*ω*</sup>) is a _P_-space if and only if the set _X_ is quasi Dedekind-finite. The space **S**(*ω*<sub>1</sub>, [*ω*<sub>1</sub>]<sup>≤*ω*</sup>) is a _P_-space if and only if *ω*<sub>1</sub> has uncountable cofinality. Every denumerable family of non-empty finite sets admits a choice function if and only if, for every infinite set _X_, either _X_ is Dedekind-infinite or **S**(_X_, [_X_]<sup>≤*ω*</sup>) is a _P_-space. If every denumerable family of non-empty subsets of the real line ℝ has a choice function, then there exists a topology 𝒯 on ℝ such that (ℝ 𝒯) is a zero-dimensional, crowded Hausdorff _P_-space. The converse implication is false in the Basic Cohen Model of **ZF**. In fact, in the Basic Cohen Model, for every infinite set _X_, the space **S**(_X_, [_X_]<sup>≤*ω*</sup>) is a _P_-space. **References** - [1] K. Keremedis, A. R. Olfati, and E. Wajch, *On P-spaces and G<sub>δ</sub>-sets in the absence of the axiom of choice*, Bull. Belg. Math. Soc. Simon Stevin 30 (2), 194–236 (2023). - [2] E. Tachtsis and E. Wajch, *Constructing crowded Hausdorff P-spaces in set theory without the axiom of choice*, submitted manuscript (2025), https://arxiv.org/abs/2510.11935
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Convergence with respect to a semitopogenous order on a complete lattice — Josef Slapal <slapal@fme.vutbr.cz>
A. Czaszar introduced the concept of a semitopogenous order as a binary relation on the power sets of a given set. We extend semitopogenous orders from power sets to arbitrary complete lattices and investigate their behaviour. In particular, we study convergence of generalized nets (upwards closed and centered subsets) with respect to a semitopogenous order and give conditions under which the convergence behaves analogously to the filter convergence in topological spaces. Separation and compactness with respect to a semitopogenous order are discussed, too.
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Dimension under Dense Linear Mappings of Function Spaces — Krzysztof Zakrzewski <kzakrze@sgh.waw.pl>
### Dimension under Dense Linear Mappings of Function Spaces All topological spaces are assumed to be Tichonoff. For a space $X$, by $dim(X)$ we denote the covering dimension of the space $X$. For a topological space $X$, let $C_p(X)$ denote the space of real continuous functions on the space *X* endowed with the pointwise convergence topology. Let $\kappa$ be an infinite cardinal number. A normal space $X$ is called strongly $\kappa$-dimensional if it is a union of $\kappa$ many closed finite dimensional spaces. For $\kappa=\omega$, one obtains the well known class of strongly countable dimensional spaces. A space is called $\kappa$-compact if it is a union of $\kappa$ many its compact subspaces. We improve results concerning invariance of dimension-like properties under transformations of function spaces from [Za]. In particular we show that for a normal, strongly $\kappa$-dimensional space $X$ and a normal, metacompact, locally $\sigma$-compact space $Y$ if there exists a continuous linear operator $T:C_p(X)\xrightarrow[]{}C_p(Y)$ with dense image, then $Y$ is strongly $\kappa$-dimensional as well. The finite-dimensional case is examined as well. We also obtain a compactification theorem that may be of independent interest: every normal strongly $\kappa$-dimensional space admits a strongly $\kappa$-dimensional $\kappa$-compactification. Recall that for a space $X$, we have $dim\,(X)=dim\,(\beta X)$ and that there exist a normal, strongly countable dimensional space without strongly countable dimensional compactification [EP,Example 5.5]. 1. [EP] R. Engelking, E. Pol, _Countable-dimensional spaces: a survey_, Dissertationes Math. 216, (1983). 1. [Za] K. Zakrzewski, _Function spaces on Corson-like compacta_, Results Math. 80, 75 (2025).
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- Plenary
Equivariant Means and Extension Properties — Natalia Jonard Pérez <nat@ciencias.unam.mx>
An $n$-mean on a topological space $X$ is a symmetric continuous operation $p:X^n\to X$ satisfying $p(x,\dots,x)=x$ for every $x\in X$. The existence of continuous means is closely related to several classical questions in topology, particularly in connection with retract theory and extension properties. In this talk, we discuss equivariant means associated with group actions on topological spaces and their connections with equivariant absolute extensors. Particular attention will be given to involutions (that is, $\mathbb Z_2$-actions) acting on spaces equipped with compatible lattice structures. We will present some existence results and applications in this setting, and explain how they relate to a classical open problem of Anderson. This is a joint work with Ananda López Poo.
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Examining Properties of Selective Divergence — Christopher Caruvana <chcaru@iu.edu>
We discuss the properties of being discretely selective and selectively highly divergent, as well as close variants. We give a variety of examples separating the notions and note their equivalence in rings of continuous real-valued functions. Some relations to hyperspaces of finite subsets are also considered.
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Function spaces on separable compact lines — Kacper Kucharski <k.kucharski6@uw.edu.pl>
A compact line is any linearly ordered compact topological space. During the talk we will provide a complete isomorphism classification of the spaces of real-valued continuous functions endowed with the topology of pointwise convergence $C_p(K)$ for separable compact lines $K$ of weight $\omega_1$, under the assumption of the Baumgartner's axiom BA. In particular, we will show that, up to linear homeomorphism, there are exactly two function spaces $C_p(K)$, for such $K$. This result should be compared with the recent work by Korpalski, Koszmider and Marciszewski in which it was proved that under the assumption of BA, whenever $K$ and $L$ are separable compact lines of weight $\omega_1$, then the Banach spaces $C(K)$ and $C(L)$ are isomorphic. We will also go over a construction of a ZFC example of a separable compact line $K$ of weight $2^{\omega}$, whose spaces of continuous functions with the pointwise convergence topology $C_p(K)$ and the weak topology $C_w(K)$ are not homeomorphic to their squares.
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Fundamental properties and characterizations of new classes of δ− β* continuous mappings in m-Polar Neutrosophic Topological Spaces — Lorenzo Affè <lorenzo.affe1@gmail.com>
We introduce and study two classes of neutrosophic continuous mappings: the neutrosophic irresolute $\delta$-$\beta^\*$-continuous mappings (NIr $\delta$-$\beta^\*$ CM) and the $\delta$-$\beta^\*$-neutrosophic contra $\delta$-$\beta^\*$-continuous mappings (NC $\delta$-$\beta^\*$ CM). We establish their fundamental properties and provide characterizations in terms of preimages of $\delta$-$\beta^\*$-open and $\delta$-$\beta^\*$-closed sets. The role of each notion related to the other is shown and analyzed through implication chains, (non-)equivalences under mild hypothesis and stability result under composition, subspaces, and products. Then an extended framework to the $m$-polar setting is shown; in particular, the definitions of the $m$-polar neutrosophic irresolute $\delta$-$\beta^\*$-continuous mappings (MPNIr $\delta$-$\beta^\*$ CM) and $m$-polar neutrosophic contra $\delta$-$\beta^\*$-continuous mappings (MPNC $\delta$-$\beta^\*$ CM) are given. Moreover, this framework shows how core properties lift to the $m$-polar case and where new phenomena arise. Also examples and counterexamples are provided in order to separate the classes and to justify and illustrate the sharpness of the obtained results.
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Further Observations on Locally Antisymmetric Spaces — Filiz YILDIZ <yfiliz@hacettepe.edu.tr>
Within the framework of asymmetry of the $T_0$-quasi-metric spaces [1], antisymmetric functions are appeared [3] as in some sense opposite to metric functions and studied [4] in detail. Following that in a previous study [2], the locality status of the $T_0$-quasi-metric spaces constructed with antisymmetric functions is described under the name local antisymmetricness. Hence, we are now in a position to ask that how local antisymmetric spaces behaves for subspaces, finite products and intersections-unions. Accordingly, some theorems and counterexamples will be presented about these observations in the context of $T_0$-quasi-metric spaces. Specifically, the question whether the images of locally antisymmetric spaces under an isometry have the same property or not, will be discussed as another problem worth examining.
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Generalizations of known cardinal inequalities for topological spaces — Ivan Gotchev <gotchevi@ccsu.edu>
In this talk we will present some new results about cardinal inequalities on topological spaces. We introduce the cardinal invariant $nu_s(X)$, the non-Urysohn number for singletons, to generalize the Urysohn separation axiom. Using this invariant, we generalize and extend some known cardinal inequalities for Urysohn spaces to all topological spaces, particularly such that involve variations of tightness and pseudocharcter. The main results pertain to upper bounds on the cardinalities of closures and $\theta$-closures of sets, and variations of the Arhangelskii-Sapirovskii inequality.
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Generalized almost disjoint families and injective Banach spaces — Chris Lambie-Hanson <lambiehanson@math.cas.cz>
We generalize the notion of almost disjoint family to the setting of arbitrary totally disconnected Hausdorff spaces. We present some results about the existence of such families on the Čech-Stone remainder of the integers. As an application, we present some modest progress concerning the open question of the injective dimension of the Banach space $$c_0$$. This is joint work with David Schrittesser.
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Hausdorff reflection preserves shape — Diego Mondéjar <diego.mondejar@cunef.edu>
We study the interaction between topological reflections and shape theory. We give general conditions under which a reflection preserves shape, showing in particular that the Hausdorff reflection induces a shape equivalence. This provides a categorical interpretation of reflections as operations that do not alter the global structure of spaces at the level of shape. Applications to inverse limits of finite $T_0$ spaces are discussed, where non-Hausdorff models retain the same shape as their Hausdorff counterparts.
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Higher Lindelöf trees — Pedro Marun <marun@math.cas.cz>
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.
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Independent sets in Abelian topological groups of prime exponent — Olga Sipacheva <ovsipa@gmail.com>
A subset $X$ of an Abelian group $G$ with zero element $0$ is said to be *independent* if, given any $n\in \mathbb N$, any pairwise distinct $x_1,\dots, x_n\in A$, and any $k_1,\dots, k_n\in \mathbb Z$, we have $k_1\cdot x_1=\dots= k_n\cdot x_n= 0$ whenever $k_1\cdot x_1 +\dots +k_n\cdot x_n=0$. In other words, $X\subset G$ is independent if the natural homomorphism $\bigoplus_{x\in X}\langle x\rangle \to G$ is injective (here $\langle x\rangle$ denotes the subgroup of $G$ generated by $x$). We say that $X$ is a *basis* of $G$ if $X$ is independent and $\langle X\rangle =G$. We consider independent subsets of Hausdorff Abelian topological groups of prime exponent $p$. It is well known that any such group $G$ is a direct sum of copies of the cyclic group $\mathbb Z/p\mathbb Z$ of order $p$ and hence can be treated as a vector space over the field $\mathbb F_p$. Therefore, $G$ has a basis $E$. Thus, on any Abelian topological group $G$ of prime exponent $p$ with basis $E$, there exists the natural topology induced by the Tychonoff product topology of $\prod_{e\in E}\langle e\rangle$. We refer to this topology as the *product topology on* $G$ *associated with* $E$. A subset $X$ of $G$ is said to be *topologically independent* if, given any $n\in \mathbb N$, any pairwise distinct $x_1,\dots, x_n\in X$, any $k_1,\dots, k_n\in \mathbb Z$, and any neighborhood $U$ of $0$, there exists a neighborhood $V$ of $0$ such that $k_1\cdot x_1, \dots, k_n\cdot x_n \in U$ whenever $k_1\cdot x_1 +\dots +k_n\cdot x_n\in V$. Clearly, any topologically independent set is independent, but the converse is not true: it is known that if $X\subset G$ is topologically independent, then the topology of $H=\langle X\rangle $ is coarser than the product topology on $H$ associated with the basis $X$ of $H$. Recall that the intersection of the kernels of continuous characters of a topological group is called the *von Neumann kernel* of $G$ and denoted by $n(G)$; a group $G$ with $n(G)= G$ is said to be *minimally almost periodic* and a group $G$ with trivial $n(G)$ is said to be *maximally almost periodic*. It is easy to see that an Abelian topological group $G$ of prime exponent is maximally almost periodic if and only if there exists a basis $E$ of $G$ such that the product topology on $G$ associated with $E$ is coarser than the original topology of $G$, i.e., $E$ is topologically independent. There exist examples of minimally almost periodic Abelian groups of any prime exponent. However, any infinite topological Abelian group of prime exponent contains an infinite maximally periodic subgroup (in other words, any such group contains an infinite topologically independent set). This is one of the main results of the report. The second main result is that any countable topological Abelian group of prime exponent has a closed discrete basis. Moreover, any countable-dimensional topological vector space over a finite field or over a complete second-countable valued field (such as $\mathbb R$ or $\mathbb C$) has a closed discrete basis. For uncountable-dimensional spaces, this is not true.
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Measuring the length of Borel hierarchies — Nick Chapman <nick.steven.chapman@gmail.com>
The class of Borel sets is one of the most fundamental structures on a topological space. Its study lies at the intersection of several areas of mathematics; in this talk, we will investigate properties of the Borel algebra from the viewpoint of descriptive set theory and topology, focusing on the length of this hierarchy on a given second-countable space $X$. The length $ord(X)$ of the hierarchy is defined as the least ordinal $\alpha$ for which every Borel subset of $X$ is $\Sigma^0_\alpha$. The exact value of this ordinal turns out to be highly malleable, and a sophisticated forcing technique was developed by Arnold Miller to produce models of set theory in which it takes on arbitrary values. We will discuss the basic building blocks of this technique, as well as sketch the nature of rank arguments that yield consistency results about assignments of $ord(X)$ to several spaces $X$ simultaneously. Time permitting, we will also delve into the speaker's recent contributions to this area, such as an extension of the framework to the study of generalized Borel hierarchies on topological spaces of uncountable weight.
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More ZFC Dowker spaces — Menachem Kojman <kojman@woobling.org>
A construction scheme of topological spaces, which generalizes M. E. Rudin's construction of a Dowker space in ZFCC, is given, and is shown to produce a proper class of Dowker spaces. A proper subclass of this class of spaces are provably collectionwise normal Dowker in ZFC alone. The theory ZFC+SSH, where SSH is Shelah's Strong Hypothesis, proves that the whole class consists of collectionwise normal Dowker spaces. Whether all members of this class are Dowker in ZFC is still open.
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Non-meager P-filters, Miller-measurability, and a question of Hrušák — Andrea Medini <andrea.medini@tuwien.ac.at>
We will discuss our recent partial answer to a question of Hrušák: if a product of filters on ω is countable dense homogeneous, then the number of factors is smaller than **p** and each factor is a non-meager P-filter. Furthermore, we will show that non-meager P-filters can be characterized as the "chunkiest" filters with respect to Miller-measurability. As a rather "quotable" corollary, we will see that the intersection of fewer that **add**(_m_<sup>0</sup>) non-meager P-filters is a non-meager P-filter, where _m_<sup>0</sup> denotes the ideal of Miller-null sets. All of these results build on an old joint paper with Kunen and Zdomskyy.
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On Set-Relatively Star-Menger Subspaces and Related Star Covering Properties — Sumit Singh <sumit@ramjas.du.ac.in>
In this paper, we study set-relatively star-Menger subspaces and their connections with classical and star covering properties. We provide characterizations of set-RSM spaces and show that the family of such subspaces forms an admissible $\sigma$-ideal. Several examples are constructed to clarify relationships with existing notions and to correct earlier claims in the literature. We also investigate preservation properties under mappings and products, and establish equivalences between relative versions of star-K-Menger, star-C-Menger, and star-K-Hurewicz properties with their corresponding set versions.
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On a Special Convergence in Cap Spaces — Meryem ATEŞ <mbiten@ankara.edu.tr>
In a topological space, Kuratowski convergence of hypernets is defined by Beer[1] and in a convergence space, Kuratowski convergence of hyperfilters is defined by Dolecki and Mynard [5]. In this study, we introduce and study upper and lower Kuratowski convergences of hyperfilters in the category Cap of convergence approach spaces and contractions. Given a convergence approach space $(X,\lambda)$, let $C_{c(\lambda)}$ denote $c(\lambda)$-closed subsets of $X$. For a hyperfilter $\mathfrak{F}$ defined on $C_{c(\lambda)}$ and $A\in C_{c(\lambda)}$ we defined: $ \lambda_{uK}\mathfrak{F}(A)=\bigvee_{x\notin A}1\oslash adh_\lambda (rdc\mathfrak{F})(x)$, $ \lambda_{lK}\mathfrak{F}(A)=\bigvee_{x\in A}1\oslash adh_\lambda (rdc\mathfrak{F}^\textit{#})(x)$ and $ \lambda_{K}\mathfrak{F}(A)=\lambda_{uK}\mathfrak{F}(A) \bigvee \lambda_{lK}\mathfrak{F}(A). $ Given an $\epsilon\in[0,\infty]$, the filter $\mathfrak{F}$ is said to be $\epsilon-$upper Kuratowski convergent (respectively $\epsilon-$lower Kuratowski convergent, respectively $\epsilon-$ Kuratowski convergent) to $A$ if $\lambda_{uK}\mathfrak{F}(A)\leq\epsilon$ (respectively $\lambda_{lK}\mathfrak{F}(A)\leq\epsilon$, respectively $\lambda_{K}\mathfrak{F}(A)\leq\epsilon$). We investigate the properties of this convergences and then obtain relations with these new notion of convergence and Fell approach structure defined by Ateş and Sagıroglu in [4]. We show that the upper Fell convergence approach structure is a non-Archimedean approach structure coarser than the upper Kuratowski convergence approach structure, but finer than the upper Fell approach structure introduced in [4]. We also obtain that if the upper Kuratowski convergence over a topological space is pretopological, then it is also topological.
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On densely defined linear continuous operators between function spaces — Arkady Leiderman <arkady@bgu.ac.il>
For any Tychonoff space $X$, let $D(X)$ denote either the space $C(X)$ of all continuous real-valued functions on $X$ or the space $C^*(X)$ of all bounded continuous real-valued functions on $X$.$\,\,\,$ We write $D_p(X)$ when $D(X)$ is endowed with the topology of pointwise convergence. In our recently published paper, A. Eysen, A. Leiderman and V. Valov, _Linear and uniformly continuous surjections between $C_p$-spaces over metrizable spaces_, Math. Slovaca, vol. 75 (2025), pp. 669--678, we obtained the following result: **Theorem.** If $T: D_{p}(X) \to D_{p}(Y)$ is a linear continuous surjection, where $X$ is a metrizable space and $Y$ is a perfectly normal space, then $Y$ inherits a given topological property $\mathcal{P}$ from $X$. A linear continuous surjection $T: E_{p}(X) \to E_{p}(Y)$ is said to be **densely defined** if $E(X)$ and $E(Y)$ are dense linear subspaces of $D_{p}(X)$ and $D_{p}(Y)$, respectively. In our talk, we establish sufficient conditions under which the above statement remains valid for a densely defined linear continuous surjection $T: E_{p}(X) \to E_{p}(Y)$. In particular, $\mathcal{P}$ can be zero-dimensionality, strong countable-dimensionality, or $\sigma$-compactness. Additionally, for arbitrary Tychonoff spaces $X$ and $Y$, assuming only that $T: E_p(X)\to E_p(Y)$ is a densely defined linear continuous operator, we show that $X\in\mathcal P$ implies $Y\in\mathcal P$ where $\mathcal P$ is the property $(\kappa)$, the strong $\sigma$-scatteredness, or the property of being a $\Delta_1$-space.
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On graph-induced betweenness — Aisling McCluskey <aisling.mccluskey@universityofgalway.ie>
Metric spaces give rise naturally to betweenness relations through the associated lens of generalised triangle (in)equalities. Examples include the usual metric betweenness of Karl Menger [1] whereby a point $c$ is said to be between points $a$ and $b$ in a metric space $(X,\rho)$ if $\rho(a,b) = \rho(a,c) + \rho(c,b)$. Another ultrametric version, contrasting sharply with Menger betweenness but aligning strongly with subcontinuum betweenness amongst hereditarily indecomposable continua, is where we declare $c$ to be between $a$ and $b$ if $\rho(a, b) = \max \{\rho(a,c), \rho(c,b)\}$. Such betweenness relations induced by metrics with values in a finite set turn out to be of interest through a natural correlation with simple graphs. We exploit this to identify when a given betweenness relation is graph-induced; namely, that edges between vertices (points of $X$) can be labelled from the set $\{1,2\}$ in such a way that the associated Menger betweenness relation from this metric (with values in the set $\{0,1,2\}$) coincides with the original betweenness relation. This is joint work with Paul Bankston (Marquette University, Wisconsin) and Steve Watson, York University, Toronto. [1] Karl Menger, Untersuchungen \"{u}ber allgemeine Metrik, Math. Ann. 100 (1928), 75--163
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On the Gromov-Hausdorff quasi-metric distance — Olivier Olela-Otafudu <olivier.olela-otafudu@ul.ac.za>
In this talk, we introduce the concept of the Gromov-Hausdorff quasi-metric distance between two quasi-metric spaces. We then use this concept to study the stability estimates of two Isbell-hulls quasi-metric spaces. Moreover, we obtain the asymmetric version of the following well-known result: the Gromov-Hausdorff distance of two hyperconvex metric spaces generated by certain subsets is less than or equal to the Gromov-Hausdorff distance of these sets.
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Ramsey spaces on trees with the successor operation — Jan Hubička <honza.hubicka@gmail.com>
Several Ramsey theorems and Ramsey spaces, such as the Milliken tree theorem and the Carlson-Simpson theorem, are naturally viewed as results about trees and their subtrees. Recently, the study of big Ramsey degrees of universal structures has led to a need for additional variants of these theorems where the notion of a subtree is modified. We discuss a general framework for proving Ramsey-type theorems on trees with finite but possibly unbounded branching and the associated Ramsey spaces. These spaces are formed by collections of infinite subtrees equipped with a topology generalizing the Ellentuck space. By verifying that these structures satisfy the abstract Ramsey space axioms, we ensure that every subset with the Baire property is Ramsey. This framework specifically incorporates the successor operation to maintain structural integrity during embeddings. This is joint work with Martin Balko, David Chodounský, Natasha Dobrinen, Matěj Konečný, Jaroslav Nešetřil, and Andy Zucker.
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Selection Principles in Cosmic Spaces — Davide Giacopello <dagiacopello@unime.it>
We introduce and investigate new selection principles involving countable networks in cosmic tychonoff spaces, namely, M-nw-selective, R-nw-selective, and H-nw-selective. These spaces represent a strengthening of both M-separability, R-separability, and H-separability, as well as the Menger, Rothberger, and Hurewicz properties. We also define and investigate two new games: the R-nw-selective game and the M-nw-selective game, which arise naturally from their corresponding selection principles. We give consistent results, and we define trivial R-, H-, and M-nw-selective spaces the cosmic ones having cardinality and weight strictly less than $\text{cov}(\mathcal{M})$, $\mathfrak{b}$, and $\mathfrak{d}$, respectively. We establish that spaces with cardinalities greater than $\text{cov}(\mathcal{M})$, $\mathfrak{b}$, and $\mathfrak{d}$ fail to possess the R-, H-, and M-nw-selective properties, respectively. Non-trivial examples, therefore, should eventually have weight greater than or equal to these small cardinals. Using forcing methods, we construct consistent countable non-trivial examples of R-nw-selective and H-nw-selective spaces. Finally, we study relations between nw-selective properties and a strong version of the HFD property.
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Some relations between two topologies on a given set — Athanasios Megaritis <acmegaritis@upatras.gr>
The study of relations between two topologies on the same set is a classical and quite old subject in General Topology with the partial order by inclusion being one of the most essential relations. Various relationships between topologies have been studied (cf. [A], [C], [D], [W]). In [M] we introduced the strongly finer relation "⊴" between two topologies on a given set X. This relation defines a new order on the family _T_(X) of all topologies on X, which is stronger than the usual subset relation. In this talk, we continue the investigation of the poset (_T_(X),⊴). In addition, we introduce some new relations between topologies on X. **References**<br> [A] J. M. Aldaz, Uniformly finer topologies, Rend. Circ. Mat. Palermo (2) 45 (1996), no. 3, 453--458.<br> [C] Vitalij A. Chatyrko, On π-compatible topologies and their special cases, Topology Appl. 374 (2025), Paper No. 109243, 10 pp.<br> [D] B. P. Dvalishvili, Bitopological spaces: theory, relations with generalized algebraic structures, and applications, North-Holland Mathematics Studies, 199. Elsevier Science B.V., Amsterdam, 2005.<br> [M] A. C. Megaritis, A new poset of topologies, Mathematica Slovaca (2026), in press.<br> [W] J. D. Weston, On the comparison of topologies, J. London Math. Soc. 32 (1957), 342--354.
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Star-Proximal Games — Jocelyn Bell <bell@hws.edu>
We introduce star variants of the proximal game in which the uniform structure is replaced by covers of a topological space and Point moves through iterated stars. The cover-star game framework decomposes the proximal game into a hierarchy of topological games that can be studied separately. These games retain several of the preservation and separation features of the proximal game while applying in settings where no uniformity is fixed or assumed.
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The Average Jones Polynomial: An Ensemble Approach to Knot Shadows via Tensors — Beomgyu Kim <posfn0319@gmail.com>
This talk introduces the Average Jones Polynomial (AJP), defined as the uniform expectation $V_{avg}(S, A) = 2^{-n} \sum_{D \in \mathcal{R}(S)} V_D(A)$, aimed to isolate the structural properties of the underlying 4-valent planar graph. We model the shadow as an uncontracted Temperley-Lieb tensor network, $\mathcal{T}(S) = \prod_{i} (a\mathbf{1} + b e_i)$. This formulation reduces the computational complexity of AJP calculations to $O(n\alpha(n)2^n)$ and maps the AJP to a finite loop-model partition function $Z_S(\delta, a, b)$. This can be utilized to evaluate some macroscopic observables (e.g., the expected number of loops). We analyze the behavior of AJP under shadow Reidemeister (SR) moves. The AJP is invariant under SR1 move, and transforms predictably under SR2 & SR3 moves. When evaluating $\Delta \mathcal{T} = \mathcal{T}(S') - \mathcal{T}(S)$, SR2 and SR3 moves appear as $e_i$ and $(e_i - e_{i+1})$ defect terms, respectively. This suggests a lower bound of required SR moves to transform one shadow into another.
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Topologies on the ring of Baire-one functions — Atasi Debray <adrpm@caluniv.ac.in>
A Baire class-one (or simply Baire-one) function $f : X \rightarrow \mathbb{R}$ is a function that can be expressed as the pointwise limit of a sequence of continuous functions on a topological space $X$. It is well known that the collection $B_1(X)$ of all Baire-one functions forms an overring of the ring $C(X)$ of continuous functions. Although $B_1(X)$ has resemblances with $C(X)$ in its algebraic behaviour, it exhibits substantial differences, particularly when endowed with topologies analogous to those commonly considered on $C(X)$. The objective of this paper is to discuss $B_1(X)$ from topological perspective and observe the behaviour of $C(X)$ as its subspace.
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Universal minimal flows of totally disconnected locally compact Polish groups — Dana Bartosova <dbartosova@ufl.edu>
Every topological group $G$ admits a universal minimal flow, $M(G),$ that is, a minimal flow that factors onto any minimal flow, that is unique up to isomorphism. Thus understanding $M(G)$ sheds light on how complicated minimal dynamics of $G$ can be. In the case of infinitely countable discrete groups, the underlying space of $M(G)$ is always the Gleason space of the Cantor cube of weight continuum, $\text{Gl}(2^{\mathfrak{c}})$, that can also be seen as the Stone space of the free completion of the free Boolean algebra on continuum many generators. Totally disconnected locally compact (TDLC) Polish non-discrete groups, in other words locally compact automorphism groups of countable structures, are topologically homeomorphic to the product of a countable discrete set and the Cantor space. In case a group $G$ is also algebraically isomorphic to a product of an infinitely countable discrete group and the Cantor group, $D\times 2^{\omega}$, then the underlying space of $M(G)$ is homeomorphic to the product $\text{Gl}(2^{\mathfrak{c}})\times 2^{\omega}$. The question is whether that is always the case. We show that the answer is positive in various scenarios, covering for instance the example of automorphism groups of finitely-branching regular countable tree. This is a joint work in progress with Andy Zucker.
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Universality theorems for mappings — DIMITRIOS GEORGIOU <georgiou@math.upatras.gr>
In this talk, we study the universality problem for the existence of universal elements in classes of continuous mappings. Especially, we present: classical results regarding universal continuous mappings and the existence of universal elements in the class of all continuous mappings from a normal space of which covering dimension is not larger than n to a fixed compact Hausdorff space Y.
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