arXiv subjects

Paul Szeptycki

Publications and source records attributed to Paul Szeptycki.

16 recordsLinked to original sources

arXiv2026

Normality in the square of the Sorgenfrey Line

We consider sets of reals $X$ endowed with the Sorgenfrey lower limit topology denoted $X[\leq]$. Przymusiński proved that if $X$ is a $Q$-set then $(X[\leq])^2$ is normal. While the converse is not in general true we consider examples of sets of the reals for which $(X[\leq])^2$ is normal or just pseudo-normal. For example, if $X$ is a $λ$ set, then $(X[\leq])^2$ is pseudo-normal but assuming CH there is an $X$ concentrated on a countable dense subset (so not a $λ$-set) but still $(X[\leq])^2$ is normal.

math.GN↗

arXiv2025

Categories of Games and their Fraïssé Theory

Relying on recent generalizations of the Fraïssé theory to a broader category-theoretic context, we study the class of abstract finite games played between two players and show the existence of an infinitetly countable game which is ultrahomogeneous and universal with respect to said class. Certain peculiarities of our game categories which clash with the usual framework found in the literature then lead us to formulate weaker category-theoretic properties which still yield a universal and ultrahomogeneous Fraïssé limit, thus further generalizing the categorical framework for a Fraïssé theory.

math.GM↗

arXiv2025

Infinitely ludic categories

Pursuing a new approach to the study of infinite games in combinatorics, we introduce the categories $\mathbf{Game}_{A}$ and $\mathbf{Game}_{B}$ and improve some classical results concerning topological games related to the duality between covering properties of $X$ and convergence properties of $\mathrm{C}_{\mathrm {p}}(X)$ by establishing the existence and key role of certain natural transformations. We then describe these ludic categories in various equivalent forms, viewing their objects as certain structured trees, presheaves, or metric spaces, and we thereby obtain their arboreal, functorial and metrical appearances. We use their metrical disguise to demonstrate a universality property of the Banach-Mazur game. The various equivalent descriptions come with underlying functors to more familiar categories which help establishing some important properties of the game categories: they are complete, cocomplete, extensive, cartesian closed, and coregular, but neither regular nor locally cartesian closed. We prove that their classes of strong epimorphisms, of regular epimorphisms, and of descent morphisms, are all distinct, and we show that these categories have weak classifiers for strong partial maps. Some of the categorical constructions have interesting game-theoretic interpretations.

math.GN↗

arXiv2025

Sequentially compact separable spaces

We consider the following variation of the Scarborough-Stone problem: Is $X^κ$ always countably compact whenever $X$ is separable and sequentially compact?

math.GN↗

arXiv2025

Q-Sets, Δ-Sets, and L-Spaces

The question whether there is a Lindelof Q-set space or Lindelof $Δ$-set space is considered. We show that J. Moore's ZFC $L$-space is not a Q-set space in ZFC and, assuming all Aronszajn trees are special, it is not a $Δ$-set space.

math.GN↗

arXiv2024

High dimensional countable compactness and ultrafilters

We define several notions of a limit point on sequences with domain a barrier in $[ω]^{<ω}$ focusing on the two dimensional case $[ω]^2$. By exploring some natural candidates, we show that countable compactness has a number of generalizations in terms of limits of high dimensional sequences and define a particular notion of $α$-countable compactness for $α\leqω_1$. We then focus on dimension 2 and compare 2-countable compactness with notions previously studied in the literature. We present a number of counterexamples showing that these classes are different. In particular assuming the existence of a Ramsey ultrafilter, a subspace of $βω$ which is doubly countably compact whose square is not countably compact, answering a question of T. Banakh, S. Dimitrova and O. Gutik. The analysis of this construction leads to some possibly new types of ultrafilters related to discrete, P-points and Ramsey ultrafilters.

math.GN↗

arXiv2024

A regular non-weakly discretely generated P-space

We construct a consistent example of a topological space $Y=X \cup \{\infty\}$ such that: 1) $Y$ is regular. 2) Every $G_δ$ subset of $Y$ is open. 3) The point $\infty$ is not isolated, but it is not in the closure of any discrete subset of $X$.

math.GN↗

arXiv2024

Semi-proximal spaces and normality

We consider the relationship between normality and semi-proximality. We give a consistent example of a first countable locally compact Dowker space that is not semi-proximal, and two ZFC examples of semi-proximal non-normal spaces. This answers a question of Nyikos. One of the examples is a subspace of $(ω+1) \times ω_1$. In contrast, we show that every normal subspace of a finite power of $ω_1$ is semi-proximal.

math.GN↗

arXiv2023

Infinite dimensional sequential compactness: Sequential compactness based on barriers

We introduce a generalization of sequential compactness using barriers on $ω$ extending naturally the notion introduced in [W. Kubiś and P. Szeptycki, On a topological Ramsey theorem, \emph{Canad. Math. Bull.}, 66 (2023), {156}--{165}]. We improve results from [C. Corral and O. Guzm{á}n and C. L{ó}pez-Callejas, High dimensional sequential compactness, \emph{Fund. Math.}] by building spaces that are $\mathcal{B}$-sequentially compact but no $\mathcal{C}$-sequentially compact when the barriers $\mathcal{B}$ and $\mathcal{C}$ satisfy certain rank assumption which turns out to be equivalent to a Katětov-order assumption. Such examples are constructed under the assumption $\mathfrak{b} =\mathfrak{c}$. We also exhibit some classes of spaces that are $\mathcal{B}$-sequentially compact for every barrier $\mathcal{B}$, including some classical classes of compact spaces from functional analysis, and as a byproduct we obtain some results on angelic spaces. Finally we introduce and compute some cardinal invariants naturally associated to barriers.

math.GN↗

arXiv2023

A Proof of the Tree Alternative Conjecture Under the Topological Minor Relation

We prove the Tree Alternative Conjecture for the topological minor relation: letting $[T]$ denote the equivalence class of $T$ under the topological minor relation we show that: $|[T]| = 1$ or $|[T]|\geq \aleph_0$ and $\forall r\in V(T)$, $|[(T,r)]| = 1$ or $|[(T,r)]|\geq \aleph_0$. In particular, by means of curtailing trees, we show that for any tree $T$ with at least one not eventually bare ray: $|[T]| \geq 2^{\aleph_0}$.

math.CO↗

arXiv2023

On $Δ$-spaces

$Δ$-spaces have been defined by a natural generalization of a classical notion of $Δ$-sets of reals to Tychonoff topological spaces; moreover, the class $Δ$ of all $Δ$-spaces consists precisely of those $X$ for which the locally convex space $C_p(X)$ is distinguished. The aim of this article is to better understand the boundaries of the class $Δ$, by presenting new examples and counter-examples. 1) We examine when trees considered as topological spaces equipped with the interval topology belong to $Δ$. In particular, we prove that no Souslin tree is a $Δ$-space. Other main results are connected with the study of 2) $Ψ$-spaces built on maximal almost disjoint families of countable sets; and 3) Ladder system spaces. It is consistent with CH that all ladder system spaces on $ω_1$ are in $Δ$. We show that in forcing extension of ZFC obtained by adding one Cohen real, there is a ladder system space on $ω_1$ which is not in $Δ$. We resolve several open problems posed in the literature.

math.GN↗

arXiv2021

Special sets of reals and weak forms of normality on Isbell-Mrówka spaces

We recall some classical results relating normality and some natural weakenings of normality in $Ψ$-spaces over almost disjoint families of branches in the Cantor tree to special sets of reals like $Q$-sets, $λ$-sets and $σ$-sets. We introduce a new class of special sets of reals which corresponds the corresponding almost disjoint family of branches being $\aleph_0$-separated. This new class fits between $λ$-sets and perfectly meager sets. We also discuss conditions for an almost disjoint family $\mathcal A$ being potentially almost-normal (pseudonormal), in the sense that $\mathcal A$ is almost-normal (pseudonormal) in some c.c.c. forcing extension.

math.GN↗

arXiv2021

On a topological Ramsey Theorem

We introduce natural strengthenings of sequential compactness called the $r$-Ramsey property for each natural number $r\geq 1$. We prove that metrizable compact spaces are $r$-Ramsey for all $r$ and give examples of compact spaces that are $r$-Ramsey but not $r+1$-Ramsey for each $r\geq 1$ (assuming CH for all $r>1$

math.GN↗

arXiv2017

$G_δ$ covers of compact spaces

We solve a long standing question due to Arhangel'skii by constructing a compact space which has a $G_δ$ cover with no continuum-sized ($G_δ$)-dense subcollection. We also prove that in a countably compact weakly Lindelöf normal space of countable tightness, every $G_δ$ cover has a $\mathfrak{c}$-sized subcollection with a $G_δ$-dense union and that in a Lindelöf space with a base of multiplicity continuum, every $G_δ$ cover has a continuum sized subcover. We finally apply our results to obtain a bound on the cardinality of homogeneous spaces which refines De La Vega's celebrated theorem on the cardinality of homogeneous compacta of countable tightness.

math.GN↗

arXiv2013

Wijsman hyperspaces of non-separable metric spaces

Given a metric space $\langle X,ρ\rangle$, consider its hyperspace of closed sets $CL(X)$ with the Wijsman topology $τ_{W(ρ)}$. It is known that $\langle{CL(X),τ_{W(ρ)}}\rangle$ is metrizable if and only if $X$ is separable and it is an open question by Di Maio and Meccariello whether this is equivalent to $\langle{CL(X),τ_{W(ρ)}}\rangle$ being normal. In this paper we prove that if the weight of $X$ is a regular uncountable cardinal and $X$ is locally separable, then $\langle{CL(X),τ_{W(ρ)}}\rangle$ is not normal. We also solve some questions by Cao, Junnilla and Moors regarding isolated points in Wijsman hyperspaces.

math.GN↗

arXiv2013

A new class of spaces with all finite powers Lindelof

We consider a new class of open covers and classes of spaces defined from them, called "iota spaces". We explore their relationship with epsilon-spaces (that is, spaces having all finite powers Lindelof) and countable network weight. An example of a hereditarily epsilon-space whose square is not hereditarily Lindelof is provided.

math.GN↗