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Translation Results for Some Star-Selection Games

Christopher Caruvana, Jared Holshouser

Abstract

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.

Translation Results for Some Star-Selection Games

Abstract

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 -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.
Paper Structure (8 sections, 29 theorems, 94 equations)

This paper contains 8 sections, 29 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Selection Games and Cover Types
  4. Star-Selection Games
  5. Sets Versus Sequences in Selection Principles
  6. Star-Selection Games in Pixley-Roy Hyperspaces
  7. Star-Selection Games in Uniform Spaces
  8. Open Questions

Key Result

Theorem 2.12

Let $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$, and $\mathcal{D}$ be collections. Suppose there are functions for each $n \in \omega$ so that Then $\mathsf{G}_{\mathrm{fin}}(\mathcal{A}, \mathcal{C}) \leq_{\mathrm{II}} \mathsf{G}_{\mathrm{fin}}(\mathcal{B}, \mathcal{D})$. If, in addition, $\overleftarrow{T}_{\mathrm{I},1} = \overleftarrow{T}_{\mathrm{I},n}$ for all $n \in \omega$, then $\mathsf

Theorems & Definitions (82)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Remark 2.9
  • Definition 2.10
  • ...and 72 more