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arXiv · 2210.00329

Translation Results for Some Star-Selection Games

Christopher Caruvana·Jared Holshouser
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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.

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Christopher Caruvana, Jared Holshouser. 2022-10-01. Translation Results for Some Star-Selection Games. https://doi.org/10.1016/j.topol.2024.108889

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