不可数集与无限线性秩序博弈

Tonatiuh Matos-Wiederhold, Luciano Salvetti
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摘要

在马修-贝克(Matthew Baker)的著作[《数学杂志》(Math. Mag. 80 (2007),第5期,第377-380页]中,出现了一个关于实数集的无限博弈,他问这个博弈能否帮助描述实数的可数子集。这个问题与巴拿赫-马祖尔博弈(Banach-Mazur Game)如何表征任意拓扑空间中的微小集合有着异曲同工之妙。在最近的一篇论文中,威尔-布赖恩(Will Brian)和史蒂文-克隆兹(Steven Clontz)证明,在贝克博弈中,当且仅当报酬集是可数集时,玩家二才有获胜策略。对此,我们给出了肯定的答案,而且,对于每一个无穷心$\kappa$,我们都构造了一个大小为$\kappa$的密集线性阶,在这个线性阶上,玩家二在所有报酬集上都有获胜策略。最后,我们提出了一些未来的研究问题,进一步强调了将布赖恩和克隆兹的描述推广到线性阶的难度。
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Uncountable sets and an infinite linear order game
An infinite game on the set of real numbers appeared in Matthew Baker's work [Math. Mag. 80 (2007), no. 5, pp. 377--380] in which he asks whether it can help characterize countable subsets of the reals. This question is in a similar spirit to how the Banach-Mazur Game characterizes meager sets in an arbitrary topological space. In a recent paper, Will Brian and Steven Clontz prove that in Baker's game, Player II has a winning strategy if and only if the payoff set is countable. They also asked if it is possible, in general linear orders, for Player II to have a winning strategy on some uncountable set. To this we give a positive answer and moreover construct, for every infinite cardinal $\kappa$, a dense linear order of size $\kappa$ on which Player II has a winning strategy on all payoff sets. We finish with some future research questions, further underlining the difficulty in generalizing the characterization of Brian and Clontz to linear orders.
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