We study the abstract Banach-Mazur game played with finitely generated structures instead of open sets. We characterize the existence of winning strategies aiming at a single countably generated structure. We also introduce the concept of weak Fraïssé classes, extending the classical Fraïssé theory, revealing its relations to our Banach-Mazur game. Finally, we exhibit connections between the universality number and the weak amalgamation property.

中文翻译：

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具有有限生成结构的游戏

我们研究了使用有限生成结构而不是开放集进行的抽象 Banach-Mazur 博弈。我们描述了针对单个可数生成结构的获胜策略的存在。我们还介绍了弱 Fraïssé 类的概念，扩展了经典的 Fraïssé 理论，揭示了它与我们的 Banach-Mazur 游戏的关系。最后，我们展示了普遍性数和弱合并属性之间的联系。