arXiv subjects

Christopher Caruvana

Publications and source records attributed to Christopher Caruvana.

15 recordsLinked to original sources

arXiv2026

On Large Covers and the Strong Closed Discrete Game

We strengthen a recent selection game equivalence proved by L. Chiozini involving spaces and their associated spaces of continuous real-valued functions. This strengthening establishes that the Rothberger game is perfect- and Markov-information dual to the strong closed discrete game on the space of real-valued continuous functions. In the process, we extend the theory of strategic equivalences for the second player in variations of the Menger and Rothberger games involving large covers while noting explicitly how large covers present serious obstacles to applying strategy translation results used by the authors in similar scenarios in previous papers. We also briefly comment on subbasic selection games introduced by D. Guerrero Sánchez and V.V. Tkachuk.

math.GN↗

arXiv2025

An Adaptation of the Vietoris Topology for Ordered Compact Sets

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.

math.GN↗

arXiv2025

Translation Results for some Selection Games with Minimal Cusco Maps

We establish relationships between various topological selection games involving the space of minimal cusco maps into the real line and the underlying domain. These connections occur across different topologies, including the topology of pointwise convergence and the topology of uniform convergence on compacta. Full and limited-information strategies are investigated. The primary games we consider are Rothberger-like games, generalized point-open games, strong fan-tightness games, Tkachuk's closed discrete selection game, and Gruenhage's W-games. We also comment on the difficulty of generalizing the given results to other classes of functions.

math.GN↗

arXiv2025

An Excursion with Divergence Properties

In this note, we compare and contrast various selective divergence properties such as the properties of being discretely selective and selectively highly divergent. We identify and incorporate a class of subsemigroups of the semigroup of strictly increasing maps from the naturals to themselves. We investigate certain implications for hyperspaces of finite subsets and characterize the closed discrete selection game on a space in terms of a particular selection game on the Vietoris hyperspace of finite subsets of that space. We also isolate some sufficient conditions on a space that guarantee that the corresponding Pixley-Roy hyperspace of finite subsets is discretely selective. We end by noting that the properties of being discretely selective and of being selectively highly divergent are equivalent in rings of continuous functions with standard topologies of uniform convergence.

math.GN↗

arXiv2024

On Traditional Menger and Rothberger Variations

We present a comprehensive report on the relationships between variations of the Menger and Rothberger selection properties with respect to $ω$-covers and $k$-covers in the most general topological setting and address the finite productivity of some of these properties. We collect various examples that separate certain properties and we carefully identify which separation axioms simplify aspects of these properties. We finish with a consolidated list of open questions focused on topological examples.

math.GN↗

arXiv2023

An Application of Descriptive Set Theory to Complex Analysis

The purpose of this paper is to prove a new general result about rings of complex analytic functions. Let $Ω$ be an arbitrary nonempty open subset of the complex plane $\mathbb C$, $\mathcal{A}(Ω)$ be the set of holomorphic functions on $Ω$ viewed as a Polish ring (not a Polish algebra over $\mathbb C$) in the usual compact open topology, let $R$ be a Polish ring and let $φ: R \to \mathcal{A}(Ω)$ be an abstract algebraic isomorphism. The main goal of this paper is to prove Theorem 36 that $φ$ is a topological isomorphism. A special result of Bers is an easy corollary. Two additional items supplement these results, viz., that $B(\mathbb{D})$, the abstract ring of bounded analytic functions on the unit disk, cannot be made into a Polish ring and that $\mathcal{M}(Ω)$, the abstract field of meromorphic functions on $Ω$, cannot be made into a Polish field.

math.CV↗

arXiv2023

The Hurewicz Property and the Vietoris Hyperspace

In this note, we characterize when the Vietoris space of compact subsets of a given space has the Hurewicz property in terms of a selection principle on the given space itself using $k$-covers and the notion of groupability introduced by Kočinac and Scheepers. We comment that the same technique establishes another equivalent condition to a space being Hurewicz in each of its finite powers. We end with some characterizations involving spaces of continuous functions and answer a question posed by Kočinac.

math.GN↗

arXiv2022

Translation Results for Some Star-Selection Games

We continue to explore the ways in which high-level topological connections arise from connections between fundamental features of the spaces, in this case focusing on star-selection principles in Pixley-Roy hyperspaces and uniform spaces. First, we find a way to write star-selection principles as ordinary selection principles, allowing us to apply our translation theorems to star-selection games. For Pixley-Roy hyperspaces, we are able to extend work of M. Sakai and connect the star-Menger/Rothberger games on the hyperspace to the $ω$-Menger/Rothberger games on the ground space. Along the way, we uncover connections between cardinal invariants. For uniform spaces, we show that the star-Menger/Rothberger game played with uniform covers is equivalent to the Menger/Rothberger game played with uniform covers, reinforcing an observation of Lj. Kočinac.

math.GN↗

arXiv2022

Selection Games with Minimal Usco Maps

We establish relationships between various topological selection games involving the space of minimal usco maps with various topologies, including the topology of pointwise convergence and the topology of uniform convergence on compact sets, and the underlying domain using full- and limited-information strategies. We also tie these relationships to analogous results related to spaces of continuous functions. The primary games we consider include Rothberger-like games, generalized point-open games, strong fan-tightness games, Tkachuk's closed discrete selection game, and Gruenhage's $W$-games.

math.GN↗

arXiv2021

On strategies for selection games related to countable dimension

Two selection games from the literature, $G_c(\mathcal O,\mathcal O)$ and $G_1(\mathcal O_{zd},\mathcal O)$, are known to characterize countable dimension among certain spaces. This paper studies their perfect- and limited-information strategies, and investigates issues related to non-equivalent characterizations of zero-dimensionality for spaces that are not both separable and metrizable. To relate results on zero-dimensional and finite-dimensional spaces, a generalization of Telgársky's proof that the point-open and finite-open games are equivalent is demonstrated.

math.GN↗

arXiv2021

Selection Games on Hyperspaces

In this paper we connect selection principles on a topological space to corresponding selection principles on one of its hyperspaces. We unify techniques and generalize theorems from the known results about selection principles for common hyperspace constructions. This includes results of Lj.D.R. Kočinac, Z. Li, and others. We use selection games to generalize selection principles and we work with strategies of various strengths for these games. The selection games we work with are primarily abstract versions of the selection principles of Rothberger, Menger, and Hurewicz type, as well as games of countable fan tightness and selective separability. The hyperspace constructions that we work with are the Vietoris and Fell topologies, both upper and full, generated by ideals of closed sets. Using a new technique we are able to extend straightforward connections between topological constructs to connections between selection games related to those constructs. This extension process works regardless of the length of the game, the kind of selection being performed, or the strength of the strategy being considered.

math.GN↗

arXiv2021

Selection Games and the Vietoris Space

We explore the connections between selection games on Hausdorff spaces and their corresponding Vietoris space of compact subsets. These considerations offer a similar relationship as the well-known relationship between $ω$-covers of $X$ and regular open covers of the finite powers of $X$. The primary utility of this method is to establish similar relationships with $k$-covers and the Vietoris space of compact subsets. Particularly, we show that some commonly studied selection principles are equivalent to a related hyperspace being Menger or Rothberger. We then apply these equivalences to correct a flawed argument in a previous paper which attempted to show that a Pawlikowski theorem is true for $k$-covers.

math.GN↗

arXiv2021

Closed Discrete Selection in the Compact Open Topology

In 2017, Tkachuk isolated the closed discrete selection property while working on problems related to function spaces [15]. In this paper we will study the closed discrete selection property and the related games and strategies on $C_k(X)$. Clontz and Holshouser showed previously that the closed discrete selection game on $C_p(X)$ is equivalent to a modification of the point-open game on $X$. In this paper we show that the closed discrete selection game on $C_k(X)$ is equivalent to a modification of the compact-open game on $X$. We also connect discrete selection properties on $C_k(X)$ to a variety of other properties on $X$, $C_k(X)$, and hyperspaces of $X$.

math.GN↗

arXiv2021

Selection Games on Continuous Functions

In this paper we study the selection principle of closed discrete selection, first researched by Tkachuk in [13] and strengthened by Clontz, Holshouser in [3], in set-open topologies on the space of continuous real-valued functions. Adapting the techniques involving point-picking games on $X$ and $C_p(X)$, the current authors showed similar equivalences in [1] involving the compact subsets of $X$ and $C_k(X)$. By pursuing a bitopological setting, we have touched upon a unifying framework which involves three basic techniques: general game duality via reflections (Clontz), general game equivalence via topological connections, and strengthening of strategies (Pawlikowski and Tkachuk). Moreover, we develop a framework which identifies topological notions to match with generalized versions of the point-open game.

math.GN↗

arXiv2017

An Extension of the Baire Property

The purpose of this paper is to define for every Polish space $X$ a class of sets, the $EBP(X)$-sets or the extended Baire property sets, to work out many properties of the $EBP(X)$-sets and to show their usefulness in analysis. For example, a proper generalization of the Pettis Theorem is proved in this context that furnishes a new automatic continuity result for Polish groups. The name extended Baire property sets is reasonable since $EBP(X)$ contains the Baire property sets $BP(X)$ and it is consistent with ZFC that the containment is proper.

math.LO↗