AI 中文总结
该研究针对收敛空间引入Menger与Rothberger选择游戏，证明拓扑收敛时还原为经典游戏，相关获胜条件对应特定覆盖性质，正则收敛空间的对应策略蕴含Alster型性质，遗传林德勒夫情形下空间为紧子集的可数并。
AI 中文摘要
我们针对收敛空间引入了Menger和Rothberger选择原则与游戏。Alice所选取的族需满足每个收敛滤子，而Bob的选择要么需保留该性质，要么仅需覆盖基础集合。当收敛为拓扑型时，这两类游戏可还原为经典游戏。要求获取L-覆盖的获胜条件满足Hurewicz与Pawlikowski刻画的类似结论；要求覆盖X的条件对应弱Menger与Rothberger游戏。我们还证明，对正则收敛空间，Bob在$\textsf G_{\fin}(\boldsymbol{\textit{C}}_L,\textit{Cov}(X))$中的获胜策略蕴含Alster型覆盖性质；在遗传林德勒夫性质下，该空间还是紧子集的可数并，且在预拓扑情形下这些紧子集是紧的。
英文摘要
We introduce Menger and Rothberger selection principles and games for convergence spaces. Alice plays families that meet every convergent filter, and Bob's selections are required either to retain this property or merely to cover the underlying set. When the convergence is topological, both games recover the classical games. The winning condition requiring an $L$-cover satisfies analogues of the Hurewicz and Pawlikowski characterizations, the condition requiring a cover of $X$ is represented by the weak Menger and Rothberger games. We also show that, for a regular convergence space, a winning strategy for Bob in $\mathsf G_{\fin}(\mathcal C_L,\Cov(X))$ implies an Alster-type covering property. Under hereditary Lindelöfness the space is moreover a countable union of compactoid subsets, which are compact in the pretopological case.
