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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.