AI 中文总结
研究de Brecht等人提出的关于清醒收敛Choquet完备空间是否域完备的问题，引入单点Choquet完备性概念，证明单点Choquet完备的T₁空间可域表示，得出收敛Choquet完备的T₁空间可域表示且清醒，简化原问题。
AI 中文摘要
de Brecht、Goubault-Larrecq、Jia和Lyu提出每个清醒的收敛Choquet完备空间是否是域完备的问题。我们引入单点Choquet完备性概念，它是收敛Choquet完备性的弱化，其中要求玩家α选择的开集有单点交集，不一定形成邻域基。我们证明每个单点Choquet完备的T₁空间是可域表示的。因此，每个收敛Choquet完备的T₁空间是可域表示的且是清醒的。在T₁情形下，上述问题中的清醒性假设是多余的，问题简化为每个收敛Choquet完备的T₁空间是否是域完备的。
英文摘要
de Brecht, Goubault-Larrecq, Jia and Lyu asked whether every sober convergence Choquet-complete space is domain-complete. We introduce the notion of singleton Choquet-completeness, a weakening of convergence Choquet-completeness in which the open sets chosen by player $α$ are required to have a singleton intersection, but not necessarily to form a neighbourhood basis. We prove that every singleton Choquet-complete $T_1$ space is domain-representable. Consequently, every convergence Choquet-complete $T_1$ space is domain-representable and hence sober. Thus, in the $T_1$ case, the sobriety assumption in the above question is redundant, and the question reduces to whether every convergence Choquet-complete $T_1$ space is domain-complete.
