Faculty publications

Permanent link for this collectionhttps://hdl.handle.net/2022/29255

Browse

Now showing 1 - 8 of 8

Selection games on continuous functions
(Elsevier B.V., 2020-07-01) Caruvana, Christopher; Holshouser, Jared
In this paper we study the selection principle of closed discrete selection, first researched by Tkachuk in [12] 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 Cp(X), the current authors showed similar equivalences in [1] involving the compact subsets of X and Ck(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.
Selection games on hyperspaces
(Elsevier B.V., 2021-08-15) Caruvana, Christopher; Holshouser, Jared
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.
Selection games and the Vietoris space
(Elsevier B.V., 2022-02-15) Caruvana, Christopher; Holshouser, Jared
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 ordinary 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 Hurewicz/Pawlikowski theorems are true for k-covers.
The Hurewicz property and the Vietoris hyperspace
(Elsevier B.V., 2023-10-01) Caruvana, Christopher
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.
Translation results for some star-selection games
(Elsevier B.V., 2024-03-24) Caruvana, Christopher; Holshouser, Jared
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.
On traditional Menger and Rothberger variations
(Universitat Politecnica de Valencia, 2024-10-01) Caruvana, Christopher; Clontz, Steven; Holshouser, Jared
We present a comprehensive report on the relationships between variations of the Menger and Rothberger selection properties with respect to w-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.
An adaptation of the Vietoris topology for ordered compact sets
(Universitat Politecnica de Valencia, 2025-12-03) Caruvana, Christopher; Holshouser, Jared
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. © 2025, Universitat Politecnica de Valencia. All rights reserved.
Exploring Relationship Dynamics of Community-Campus Partnerships
(2023-04-21) Coppola, Angela M.; Shukla, Anubhuti; Clark-Wiltz, Meredith; Bii, Kiplang’at (Chip); Weaver, Laura
There is a need for faculty and institutions to establish a stronger knowledge-base and more supportive structures that ensure community engagement initiatives are effective while also being equitable and authentic for all partners. The purpose of this project was to begin exploring partnership values to identify descriptors that serve as enacting values in which partners can act upon to set up a strong foundation and identify partnership strengths and areas for improvement. The authors explored existing research on how to foster specific partnership values, finding 55 articles and identifying 75 descriptors about enacting partnership values. Five categories of enacting values were identified from a thematic analysis of the descriptors: partnership foundation building, decision-making and flow of input, power distribution, mutual engagement, and multidimensional and multi-beneficial outcomes. The broader perceived values, types of relationships, and descriptors within categories were used to create a tool for reflection, discussion, and advocacy for partnership growth and development. This tool can assist those in community-campus partnerships to create strong partnerships and assess their needs to facilitate discussions between partners, and with institutions/organizations, and external groups who can provide professional development, resources, or assistance.

Browse

Recent Submissions